Does my house just not have a plumbing cleanout?
The cleanout would typically be at the base of the main plumbing stack and/or just inside or outside the home. When located outside, it's common for them to get buried over with landscaping. You should also have one every 100' to the connection with the municipal sewer. When located inside, they may be in the wall with a removable cover on the drywall instead of a large plug in the floor.

That said, I believe most plumbers would just pull the toilet and snake it from there. You know the clog is downstream of that toilet, and you know that all drain connections from the toilet down should be at least 3" to run a large snake down the line.

Here a bit more of the requirements for a cleanout from the 2012 plumbing code which I doubt have changed much since your home was built (mine in the early 1980s has the cleanout at the base of the stack and another in the front yard). Note there are lots of exceptions, like not requiring cleanouts for anything above the bottom floor of the residence that can be found in the source link:


  2012 International Plumbing Code - Cleanout requirements:
  
  IPC - Section 708 Cleanouts
  
  IPC - 708.1 Scope. This section shall govern the size, location,
  installation and maintenance of drainage pipe cleanouts.
  
  IPC - 708.2 Cleanout plugs. Cleanout plugs shall be brass or plastic,
  or other approved materials. Brass cleanout plugs shall be utilized
  with metallic drain, waste and vent piping only and shall conform to
  ASTM A 74, ASME A112.3.1 or ASME A112.36.2M. Cleanouts with
  plate-style access covers shall be fitted with corrosion-resisting
  fasteners. Plastic cleanout plugs shall conform to the requirements of
  Section 702.4. Plugs shall have raised square or countersunk square
  heads. Countersunk heads shall be installed where raised heads are a
  trip hazard. Cleanout plugs with borosilicate glass systems shall be
  of borosilicate glass.
  
  IPC - 708.3 Where required. Cleanouts shall be located in accordance
  with Sections 708.3.1 through 708.3.6.
  
  IPC - 708.3.1 Horizontal drains within buildings. All horizontal
  drains shall be provided with cleanouts located not more than 100 feet
  (30 480 mm) apart.
  
  IPC - 708.3.2 Building sewers. Building sewers shall be provided with
  cleanouts located not more than 100 feet (30 480 mm) apart, measured
  from the upstream entrance of the cleanout. For building sewers 8
  inches (203 mm) and larger, manholes shall be provided and located not
  more than 200 feet (60 960 mm) from the junction of the building drain
  and building sewer, at each change in direction and at intervals of
  not more than 400 feet (122 m) apart. Manholes and manhole covers
  shall be of an approved type.
  
  IPC - 708.3.3 Changes of direction. Cleanouts shall be installed at
  each change of direction greater than 45 degrees (0.79 rad) in the
  building sewer, building drain and horizontal waste or soil lines.
  Where more than one change of direction occurs in a run of piping,
  only one cleanout shall be required for each 40 feet (12 192 mm) of
  developed length of the drainage piping.
  
  IPC - 708.3.4 Base of stack. A cleanout shall be provided at the base
  of each waste or soil stack.
  
  IPC - 708.3.5 Building drain and building sewer junction. There shall
  be a cleanout near the junction of the building drain and the building
  sewer. The cleanout shall be either inside or outside the building
  wall and shall be brought up to the finished ground level or to the
  basement floor level. An approved two-way cleanout is allowed to be
  used at this location to serve as a required cleanout for both the
  building drain and building sewer. The cleanout at the junction of the
  building drain and building sewer shall not be required if the
  cleanout on a 3-inch (76 mm) or larger diameter soil stack is located
  within a developed length of 10 feet (3048 mm) of the building drain
  and building sewer connection. The minimum size of the cleanout at the
  junction of the building drain and building sewer shall comply with
  Section 708.7.
  
  IPC - 708.3.6 Manholes. Manholes serving a building drain shall have
  secured gas-tight covers and shall be located in accordance with
  Section 708.3.2.
  
  IPC - 708.4 Concealed piping. Cleanouts on concealed piping or piping
  under a floor slab or in a crawl space of less than 24 inches (610 mm)
  in height or a plenum shall be extended through and terminate flush
  with the finished wall, floor or ground surface or shall be extended
  to the outside of the building. Cleanout plugs shall not be covered
  with cement, plaster or any other permanent finish material. Where it
  is necessary to conceal a cleanout or to terminate a cleanout in an
  area subject to vehicular traffic, the covering plate, access door or
  cleanout shall be of an approved type designed and installed for this
  purpose.
  
  IPC - 708.5 Opening direction. Every cleanout shall be installed to
  open to allow cleaning in the direction of the flow of the drainage
  pipe or at right angles thereto.
  
  IPC - 708.6 Prohibited installation. Cleanout openings shall not be
  utilized for the installation of new fixtures, except where approved
  and where another cleanout of equal access and capacity is provided.
  
  IPC - 708.7 Minimum size. Cleanouts shall be the same nominal size as
  the pipe they serve up to 4 inches (102 mm). For pipes larger than 4
  inches (102 mm) nominal size, the size of the cleanout shall be not
  less than 4 inches (102 mm).


 Source: http://plumbingengineer.com/content/code-requirements-drainage-system-cleanouts