What is the hybridization of chlorine in vinyl chloride?
Disclaimer: An important point to remember is, that hybridisation always follows the molecular structure, it is never the cause for a certain structure. As such, hybridisation is an interpretation tool, nothing more.



As a rule of thumb, terminal atoms (excluding hydrogen) are almost always (maximally) approximately sp hybridised. While other hybridisation schemes can be applied, they are usually not a good representation.*

One popular example is water. In many texts the central oxygen is described as having (approximately) four sp3 orbitals, which essentially makes the lone pairs equivalent. That view, however, is not in agreement with the photoelectron spectrum, which clearly shows that the lone pairs are not equivalent. At this point I'd like to refer you to Michael Laing's article: "No rabbit ears on water. The structure of the water molecule: What should we tell the students?" (J. Chem. Educ. 1987, 64 (2), 124.).

The same principle described there applies to terminal atoms. Because of the local $C_\mathrm{\infty}$ symmetry the notion of three equivalent lone pairs is likely to be false. The most likely (main resonance contributor) structure for these atoms are (if hybridisation is at all feasible, i.e. mostly in the second period) one sp orbital forming the bond, one sp lone pair, two (perpendicular) p lone pairs.

In this particular example, I have calculated the molecule on the DF-BP86/def2-SVP level of theory and analysed it with the natural bond orbital theory (NBO6). Below you find the localised orbitals which best fit with the common Lewis structure of the molecule.
I have ordered the orbitals starting with the carbon-carbon &sigma;- and carbon-chlorine &sigma;-bonds on the bottom. Following that is the carbon-carbon &pi;-bond. Continuing with the three carbon-hydrogen &sigma;-bonds, and finally ending with the three chlorine lone pairs. The occupied orbitals are in blue and orange, while the corresponding virtual orbitals are next to it in red and yellow. The detailed analysis in numbers is at the end of the post.



In ast's answer there is the claim that chlorine must be sp2 hybridised because of a resonance contributor. While this explanation is extremely tempting because it easy to understand, you cannot judge from the possibility of resonance to the electronic structure. As another general rule of thumb you can remember: the lesser the orbitals are hybridised, the more likely the structure is.
The calculation also provides an analysis in terms of natural resonance theory. On the DF-BP86/def2-SVP level the contribution of the second configuration to the total electronic structure is only 6%. The main configuration contributes with about 90%, other configurations (ionic) make up the remaining 4%.

In summary, I'm afraid, but your book is incorrect, or at least incomplete. The description with two equivalent sp2 lone pairs is unnecessary complicated and does not accurately reproduce the electronic structure.



Truncated results of the NBO analysis (skipping core orbitals)


     (Occupancy)   Bond orbital / Hybrids
 ------------------ Lewis ------------------------------------------------------
   8. (1.99378) LP ( 1)Cl  6            s( 81.55%)p 0.23( 18.44%)d 0.00(  0.01%)
   9. (1.97199) LP ( 2)Cl  6            s(  0.18%)p99.99( 99.79%)d 0.18(  0.03%)
  10. (1.90237) LP ( 3)Cl  6            s(  0.00%)p 1.00( 99.95%)d 0.00(  0.05%)
  11. (1.99642) BD ( 1) C  1- C  2
               ( 48.61%)   0.6972* C  1 s(  0.00%)p 1.00( 99.94%)d 0.00(  0.06%)
               ( 51.39%)   0.7169* C  2 s(  0.00%)p 1.00( 99.95%)d 0.00(  0.05%)
  12. (1.99535) BD ( 2) C  1- C  2
               ( 49.24%)   0.7017* C  1 s( 40.16%)p 1.49( 59.76%)d 0.00(  0.08%)
               ( 50.76%)   0.7124* C  2 s( 44.47%)p 1.25( 55.47%)d 0.00(  0.06%)
  13. (1.96941) BD ( 1) C  1- H  3
               ( 61.15%)   0.7820* C  1 s( 29.81%)p 2.35( 70.15%)d 0.00(  0.04%)
               ( 38.85%)   0.6233* H  3 s( 99.91%)p 0.00(  0.09%)
  14. (1.98105) BD ( 1) C  1- H  4
               ( 61.33%)   0.7832* C  1 s( 30.09%)p 2.32( 69.87%)d 0.00(  0.04%)
               ( 38.67%)   0.6218* H  4 s( 99.91%)p 0.00(  0.09%)
  15. (1.98492) BD ( 1) C  2- H  5
               ( 62.36%)   0.7897* C  2 s( 32.79%)p 2.05( 67.18%)d 0.00(  0.03%)
               ( 37.64%)   0.6135* H  5 s( 99.89%)p 0.00(  0.11%)
  16. (1.99056) BD ( 1) C  2-Cl  6
               ( 43.14%)   0.6568* C  2 s( 22.75%)p 3.39( 77.00%)d 0.01(  0.25%)
               ( 56.86%)   0.7541*Cl  6 s( 18.31%)p 4.43( 81.12%)d 0.03(  0.57%)
 ---------------- non-Lewis ----------------------------------------------------
  17. (0.09516) BD*( 1) C  1- C  2
               ( 51.39%)   0.7169* C  1 s(  0.00%)p 1.00( 99.94%)d 0.00(  0.06%)
               ( 48.61%)  -0.6972* C  2 s(  0.00%)p 1.00( 99.95%)d 0.00(  0.05%)
  18. (0.01232) BD*( 2) C  1- C  2
               ( 50.76%)   0.7124* C  1 s( 40.16%)p 1.49( 59.76%)d 0.00(  0.08%)
               ( 49.24%)  -0.7017* C  2 s( 44.47%)p 1.25( 55.47%)d 0.00(  0.06%)
  19. (0.00909) BD*( 1) C  1- H  3
               ( 38.85%)   0.6233* C  1 s( 29.81%)p 2.35( 70.15%)d 0.00(  0.04%)
               ( 61.15%)  -0.7820* H  3 s( 99.91%)p 0.00(  0.09%)
  20. (0.01032) BD*( 1) C  1- H  4
               ( 38.67%)   0.6218* C  1 s( 30.09%)p 2.32( 69.87%)d 0.00(  0.04%)
               ( 61.33%)  -0.7832* H  4 s( 99.91%)p 0.00(  0.09%)
  21. (0.02582) BD*( 1) C  2- H  5
               ( 37.64%)   0.6135* C  2 s( 32.79%)p 2.05( 67.18%)d 0.00(  0.03%)
               ( 62.36%)  -0.7897* H  5 s( 99.89%)p 0.00(  0.11%)
  22. (0.03245) BD*( 1) C  2-Cl  6
               ( 56.86%)   0.7541* C  2 s( 22.75%)p 3.39( 77.00%)d 0.01(  0.25%)
               ( 43.14%)  -0.6568*Cl  6 s( 18.31%)p 4.43( 81.12%)d 0.03(  0.57%)


Summary of Natural Population Analysis


                                     Natural Population
             Natural    ---------------------------------------------
  Atom No    Charge        Core      Valence    Rydberg      Total
 --------------------------------------------------------------------
    C  1   -0.43992      1.99995     4.42868    0.01129     6.43992
    C  2   -0.22801      1.99996     4.20978    0.01827     6.22801
    H  3    0.22758      0.00000     0.76996    0.00246     0.77242
    H  4    0.22498      0.00000     0.77165    0.00337     0.77502
    H  5    0.23463      0.00000     0.76237    0.00300     0.76537
   Cl  6   -0.01926      9.99998     7.00353    0.01575    17.01926
 ====================================================================
 * Total *  0.00000     13.99989    17.94598    0.05414    32.00000


Optimised Geometry DF-BP86/def2-SVP


6
symmetry cs ( E = -538.027155015 A.U. )
C        1.309538928      1.048660302      0.000000000
C        0.000000000      0.759346920      0.000000000
H        1.630332304      2.102388803      0.000000000
H        2.083175437      0.265507033      0.000000000
H       -0.793639761      1.523771533      0.000000000
Cl      -0.633947150     -0.867041806      0.000000000




* It is also important to understand, that atoms are never hybridised, only orbitals are. The presence of an sp hybrid orbital does not exclude the presence of a sp3 hybrid orbital.