It's your world. It's also technically possible. So I guess the answer is yes.
What do you need for it to happen?
Most plastics, even today, are derived from mineral oil. Which is a shame, really, because there are other sources perfectly suitable that would cause a lot less problems, but I digress.
So you want a thriving petrochemical industry, and you want a need that accelerates the development of plastics for use in weapons manufacturing.
For the need, all you have to do is pick a country, and someone who decides it might be a good idea.
Machining wood to provide stocks and grips is perfectly feasible, but pressure casting of molten plastic should be more efficient. So assuming your weapons manufacturer simply found that using plastic would be cheaper, given the tall orders he got from the government, it's all a question of wanting to do it.
Let it happen in germany, for their derire for the most advanced stuff you could think of. You could also use the US, for their large oil reserves at the time.
The rest should move along just fine.
Add into the story the fact that wooden stocks and grips need more maintenance (especially in wet conditions which you get at times in a battle) than plastic does, and you should be fine.
But in the end it really just boils down to:
- Is it possible? Yes. See above.
- Do I want it? Yes. Otherwise you wouldn't have asked the question.
- Can I make it credible? Yes. See above, if you really have to, or just ignore it. After all, it's your world!