Well, the answer to this question is: we don't actually know much about the sexual practises of prehistoric humans because we weren't there, and given the diversity of human sexual norms it is probable there were a number of different systems in place depending on the group in question. Some human communities in the world today practise "wife-sharing", where wives are wives to the whole group. Some groups today practise strict monogamy. Some groups are in-between. So prehistoric humans were probably somewhat similar.

But one likely scenario is "polygyny", which is "one man, many or multiple women". Some mathematical models suggest that STD outbreaks in relatively small polygynous communities are negligible, and the community can bounce back. However in larger polygynous populations STDs become endemic and difficult to shake. So it is possible that STD outbreaks in larger, more agricultural, populations actually triggered a shift towards monogamy. That's a bit later than we're talking about, though.

But biologically speaking there is absolutely no evidence to suggest that early humans couldn't contract what we would call "STDs". It's likely that they didn't contract the same diseases we can pick up today, simply because it's been such a long time that their diseases will probably have become extinct or evolved into different diseases. But an "STD" is just a disease that can be and ordinarily is transmitted sexually, and it doesn't have to be exclusively spread through sexual contact either.

Most of what we know about prehistoric people comes from their physical records, such as bones or artefacts or paintings etc. So you can see the problem: many diseases don't leave a "record" on the bones, which are generally the only things we can find and look at. And even if some diseases do, the bones of the people who caught them might not be preserved. Prehistory is fun like that.

But we can be very confident that early humans would have contracted diseases through sexual contact even if we don't know what they suffered from because many diseases can be transmitted sexually, including diseases which we do not call "sexually transmitted" (e.g. meningitis).

And the STDs we have today and know from the historical record had to have come from somewhere, that is, they didn't pop up out of the aether some time in the 1350s (okay, technically they might have spontaneously mutated from a non-pathogenic bacterial species but at least some probably have a longer history than that, and even if none of the currently extant STDs have a storied pathogenic history they at least illustrate the point that these diseases can exist and could have existed). So to sum up, yes they could probably contract diseases we would call STDs, but we don't know how prevalent they were or what they were necessarily.