Why aren't there any trajectories for the Viking missions in JPL Horizons' database?

The real short version is that Horizons is maintained by a single person who has other assignments at NASA, and only adds missions as they are requested by the public, or newly launched. Older missions that were still active ~20 years ago. There are archives where this older data can be pulled, just not web based.

There are some other reasons for this. One of the more interesting is that each mission has a number associated with it called the NAIF id, some of which were reused, for reasons I don't understand. These NAIF ids are used in the Horizons files, because Horizons is built on top of SPICE, and they are used in the SPICE files. Just to give one example, Viking 2 and Deep Space 1 share the code of -30.

The single biggest exception to this is the Apollo missions, which I assume are in there because of the huge interest in those missions. Viking just isn't that interesting these days, because we have much better data of Mars than anything that came from Viking, except for maybe the landers, which don't move and don't require SPICE files.

You can find the data for many of these older missions, including the Viking Orbiters, in the form of the SPICE kernels, at NASA's NAIF archives.