How to perform a relaxed scan in GAMESS?
Unfortunately a feature like ModRedundant in Gaussian, which one can use to perform relaxed scans, is not available in GAMESS. Even the possibility to perform a rigid scan seems a bit complicated. You can request a calculation with RUNTYP=surface and you have to specify a $SURF group. See example 29 of the tests, which you should find in path/to/gamess/tests/standard/exam29.inp.

A relaxed scan is a bit harder to perform, and only straight forward for varying distances. You have to perform a series of single point calculations, with partially frozen coordinates. I describe the general procedure of doing that in How do I perform a partial optimisation in GAMESS?

Here I will outline it for your problem, based on what you can read there.

Before we do that however, you should be aware, that there is a python script by Andrew Hoos on github. I did not have the time to test it. It is quite old, and I cannot predict how the development of GAMESS will influence the functioning of rscan.

I assume you have read and understood the linked post, so in an effort to safe some space, I use a more automated approach here.

First, set up the general file, which you want to use for a check run. I have used your geometry.


 $CONTRL
   EXETYP=CHECK
   SCFTYP=RHF RUNTYP=ENERGY
   COORD=UNIQUE   ICHARG=1   MULT=1   NZVAR=15   $END
 $BASIS
   GBASIS=STO    NGAUSS=3   $END
 $ZMAT
   AUTO=.TRUE.   DLC=.TRUE.
   NONVDW(1)=1,5, 2,5
   $END
 $GUESS
   GUESS=HUCKEL    $END
 $DATA
   H5O2
   C1
   OXYGEN      8.0      0.086655000         0.133386000        -0.150088000
   OXYGEN      8.0     -1.636963000         0.213966000        -1.809353000
   HYDROGEN    1.0     -0.149749000         0.260762000         0.763793000
   HYDROGEN    1.0      0.802607000        -0.484925000        -0.261350000
   HYDROGEN    1.0     -0.423645000         0.140377000        -0.629197000
   HYDROGEN    1.0     -2.313930000        -0.430164000        -1.970372000
   HYDROGEN    1.0     -1.720592000         0.938467000        -2.416119000
   $END


It is again essential that you include NONVDW descriptions. Since the proton will move in between the two water molecules, I have included both pairs. Note that the 1-2 distance will not be included in the list of internal coordinates (even if you specify it directly). You can use NONVDW(1)=1,2 instead though.

Now that you have checked everything you can run the first optimisation:


 $CONTRL
   SCFTYP=RHF   RUNTYP=OPTIMIZE    COORD=UNIQUE
   ICHARG=1   MULT=1   NZVAR=15     $END
 $BASIS
   GBASIS=STO    NGAUSS=3   $END
 $ZMAT
   AUTO=.TRUE.   DLC=.TRUE.
   NONVDW(1)=1,5, 2,5
   IFZMAT(1)=1,1,2,     1,1,5
   FValue(1)=2.3938489, 0.700
   $END
 $GUESS
   GUESS=HUCKEL   $END
 $DATA
   H5O2
   C1
   OXYGEN      8.0      0.086655000         0.133386000        -0.150088000
   OXYGEN      8.0     -1.636963000         0.213966000        -1.809353000
   HYDROGEN    1.0     -0.149749000         0.260762000         0.763793000
   HYDROGEN    1.0      0.802607000        -0.484925000        -0.261350000
   HYDROGEN    1.0     -0.423645000         0.140377000        -0.629197000
   HYDROGEN    1.0     -2.313930000        -0.430164000        -1.970372000
   HYDROGEN    1.0     -1.720592000         0.938467000        -2.416119000
   $END


Note that I am keeping the automagic creation of the coordinates this time. To be on the safer side, you should probably follow more closely the other post. In any case, we are now ready to continue. Well use the optimised geometry of the last run, as well as the orbitals, which can be found in the auxiliary output.


 $CONTRL 
   SCFTYP=RHF    RUNTYP=OPTIMIZE    COORD=UNIQUE
   ICHARG=1   MULT=1   NZVAR=15     $END
 $BASIS  
   GBASIS=STO    NGAUSS=3   $END
 $ZMAT
   AUTO=.TRUE.   DLC=.TRUE.
   NONVDW(1)=1,5, 2,5
   IFZMAT(1)=1,1,2,     1,1,5
   FValue(1)=2.3938489, 0.800
   $END
 $GUESS  
   GUESS=MOREAD 
   NORB=15
   $END

! ----- RESULTS FROM SUCCESSFUL RHF      GEOMETRY SEARCH -----
! ----- COORDS, ORBS, GRADIENT, AND APPROX. HESSIAN -----
!  COORDINATES OF SYMMETRY UNIQUE ATOMS (ANGS)
!      ATOM  CHARGE       X              Y              Z
! ------------------------------------------------------------
 $DATA
   O2H5
   C1
   OXYGEN      8.0   0.0758752612   0.1029703596  -0.1380388056
   OXYGEN      8.0  -1.6329486914   0.2211188064  -1.8103049937
   HYDROGEN    1.0  -0.0725810430   0.4747088980   0.7504261439
   HYDROGEN    1.0   0.9330501410  -0.3174832331  -0.3330916306
   HYDROGEN    1.0  -0.4237766906   0.1376065810  -0.6270666776
   HYDROGEN    1.0  -2.3424944360  -0.4268887357  -2.0060894609
   HYDROGEN    1.0  -1.7921347121   0.9257714517  -2.4735439582
   $END

! --- OPTIMIZED RHF      MO-S --- GENERATED AT Fri Dec 15 00:48:25 2017
! E=     -150.1620919294, E(NUC)=   47.9559336673
 $VEC   
 1  1 9.93613099E-01 3.20702947E-02-2.24634863E-03 1.56904217E-04-2.19957003E-03
 1  2 3.13313084E-04 9.04941985E-04 6.59436960E-04-4.52587984E-05 6.45901704E-04
 1  3-4.70970670E-03-4.70987324E-03-9.86284464E-03 1.88174472E-04 1.88109591E-04
 2  1-5.80880573E-04 1.71794788E-03-8.47977388E-04 5.87852394E-05-8.29926100E-04
 2  2 9.94111746E-01 2.64448946E-02-2.47981701E-03 1.59721737E-04-2.45660777E-03
 2  3 1.64987868E-04 1.65066081E-04-3.05915593E-03-5.57763329E-03-5.57725393E-03
 3  1 2.41238581E-01-8.54557950E-01 8.35552605E-02-5.83631489E-03 8.18285646E-02
 3  2 7.39936297E-03-2.26953665E-02-1.91551065E-03 1.44103019E-04-1.84324895E-03
 3  3-7.30179599E-02-7.30300125E-02-1.26560023E-01-2.80500756E-05-2.84225764E-05
 4  1-1.47611593E-02 6.57491207E-02 2.15954260E-02-1.48570875E-03 2.11365513E-02
 4  2 2.35829657E-01-8.53953087E-01 6.86900515E-02-4.36681579E-03 6.81986435E-02
 4  3 1.56173985E-02 1.56167812E-02-3.51168589E-02-1.45207104E-01-1.45193326E-01
 5  1 2.65621341E-02-1.38559271E-01-5.11829955E-01 3.54136523E-02-5.00838599E-01
 5  2 1.64839489E-02-6.35695788E-02 4.44897012E-02-3.02940337E-03 4.36454714E-02
 5  3-2.12387862E-01-2.12451288E-01 3.07944883E-01-3.22077160E-02-3.22120362E-02
 6  1-6.42384605E-06 3.19756766E-05-4.30678730E-01 3.39316521E-01 4.64142719E-01
 6  2-8.23546629E-06 3.50155053E-05-4.64034770E-03 3.65246668E-03 4.98286518E-03
 6  3 3.26546905E-01-3.26522265E-01-2.41992633E-05-2.15894395E-05 1.43171984E-05
 7  1 3.03726648E-05-1.46010812E-04 3.61316021E-01 8.78725612E-01-3.07129752E-01
 7  2-9.63734293E-07 4.12738444E-06 1.19803044E-02 2.92946887E-02-1.01831260E-02
 7  3 3.50564742E-05 4.37481046E-05 1.83831688E-05-2.02321039E-02 2.02282616E-02
 8  1 3.91710932E-06-1.95703405E-05 1.97311245E-02 4.80455996E-02-1.67388085E-02
 8  2 5.95251643E-06-3.08145320E-05-2.27533068E-01-5.59223458E-01 1.93232191E-01
 8  3 3.64914685E-05-2.06253221E-05 8.95712681E-08 4.24180458E-01-4.24177941E-01
 9  1-2.23062153E-02 1.54500963E-01-1.24786038E-01 8.66527729E-03-1.22090919E-01
 9  2 8.76656449E-02-4.57576664E-01-5.50893665E-01 3.63552632E-02-5.43466463E-01
 9  3-3.91196174E-02-3.91748912E-02-1.19065907E-01 2.64674392E-01 2.64677791E-01
10  1 1.09047481E-04-7.57403559E-04-4.23084857E-03 3.61227555E-03 5.66486586E-03
10  2 1.69891575E-04-8.14724604E-04 6.04758690E-01-4.68585751E-01-6.43938341E-01
10  3 5.24314248E-03-4.87589514E-03 5.88487922E-04 2.55282495E-04 2.68854270E-04
11  1 1.54833774E-01-1.06847947E+00-2.73466865E-01 1.87755601E-02-2.67446840E-01
11  2 2.46273283E-02-1.43176172E-01-9.48992300E-03 8.45045780E-04-8.70879842E-03
11  3 7.99564096E-01 7.99664119E-01 2.53819180E-01 7.71090350E-02 7.71448689E-02
12  1 5.72763567E-06-2.78066490E-05 5.58361863E-01-4.39877926E-01-6.01681240E-01
12  2-9.45283650E-06 6.04268124E-05-9.09590832E-05 2.17244079E-05 3.25012306E-05
12  3 8.69427097E-01-8.69318789E-01-3.23352070E-05-6.93390818E-05-3.27231202E-05
13  1 4.13030403E-03-7.72647532E-02 1.79424923E-01-1.24368684E-02 1.75491985E-01
13  2 1.34789464E-01-8.74151452E-01 4.71611018E-01-3.07542286E-02 4.66703957E-01
13  3-1.13602815E-01-1.13734341E-01 2.65565976E-01 7.69534468E-01 7.69759188E-01
14  1-1.26314109E-06 1.52361151E-05-7.19364333E-04-1.76212662E-03 5.75477468E-04
14  2 2.07150162E-05-1.49811355E-04 3.43071888E-01 8.42940764E-01-2.91210891E-01
14  3 2.75400846E-05-1.38879856E-05-3.40876355E-05 8.31340062E-01-8.31078482E-01
15  1-8.63113810E-02 9.44284502E-01-7.40065924E-01 5.12577143E-02-7.24284153E-01
15  2-1.65007257E-02 6.66544153E-02 2.89063586E-01-1.96228027E-02 2.83830925E-01
15  3 2.15524124E-01 2.15552519E-01-1.69321183E+00 1.62283414E-01 1.62340378E-01
 $END   


And you'll have to do that for every step of you scan. If you are a bit talented with scripting, then you can surely automate the process.

I did not run the whole scan, but here are the energies of the ones I completed:


OH = 0.7   #1       NSERCH= 13     ENERGY=    -150.1620919
OH = 0.8   #2       NSERCH=  4     ENERGY=    -150.2905386
OH = 0.9   #3       NSERCH=  4     ENERGY=    -150.3504096
OH = 1.0   #4       NSERCH=  5     ENERGY=    -150.3755229
OH = 1.1   #5       NSERCH= 60     ENERGY=    -150.3875894
OH = 1.2   #6       NSERCH=  9     ENERGY=    -150.3898837
OH = 1.3   #7       NSERCH= 20     ENERGY=    -150.3872606
OH = 1.4   #8       NSERCH= 36     ENERGY=    -150.3774125