Help! This FTM knows nothing about how play dates work.

I recently became a SAHM and I have a 2.5 yr old daughter who currently goes to preschool. I just had a conference call with her teachers about her progress report and they suggested maybe I can consider play dates with some of her classmates to enhance her peer socialization skills. I guess I am open to the idea but I have no idea how play dates work. I have no friends (let alone mom friends), so no one experience whatsoever. I am a socially anxious person, a home body, and pretty introverted, so I keep thinking it would be a nuisance and a burden to cold call these moms asking to hang out with the kids, lol. Any thoughts? What if they don’t really want to since a lot of them seem to have siblings at home already? Am I just overthinking? I did meet a sweet mom of a boy classmate at the local library and we said hi. But he has an older sister. In cold weathers like this, where do play dates happen?