The verb in the sentence here is indeed vivre. Comments by Catonic, Stéphane, Graffito and SF_user are all true in a way.
So in this context, il vit des femmes indeed means he lives off (some - those of the neighbourhood for example, acquaintances of his...) women. This is not the same as il vit de femmes, which does seem weird.
As a French, the immediate meaning I get from il vit de femmes is "he lives on women", that is, he satiates basic physiological needs (either he eats them, or has lustful, indiscriminate sex with them for example, which galanterie forbids). The correct English translation of Camus' sentence would be in this case "he lives off women" as SF_user points out. A way to paraphrase that in French would be il est entretenu par des femmes.
Another nuance in using des femmes instead of de femmes is that Camus is not specific about which particular women, but not entirely unspecific altogether: he is not saying he is living off womankind. He means the protagonist:
- is living off some women - he does not really care which one in particular
- yet he is not living off any woman in the world: it has to be women he knows, neighbours, relatives...
Please note the importance of context here: the reader at this point has an idea of which women Camus is talking about. It is perfectly correct and not weird to say : il vit de petits larcins (he lives off petty crime). In this case, I understand he is not directly sating basic needs, but that he is involved in a fishy business to get some money somewhere. On the contrary, if someone said il vit des petits larcins, it would seem weird: which ones? It would translate as "he lives off the petty crimes", and would seem strange.
Last, you are correct: il vit may be 3rd person singular passé simple for voir des femmes.
voir des femmes would then mean "he dates some women, he meets some women on a regular basis, and rather casually".
Here, both context and tense are crucially important.
- In French, passé simple denotes a brief action, well situated in time. For example, we could use it to underline an extraordinary situation, such as il vit une très belle femme or il vit un très gros chien. So using voir meaning "to date, on a casual, somewhat lengthy basis" with passé simple does not fit. We would use imparfait in this case, and say, "il voyait des femmes", all the more so because passé simple belongs to the written language and is very formal, whereas voir des femmes is quite casual.