You ask the Mom.
Me, I don't mind if the Moms drop their kid off. Nor do I mind if they stay.
So, there is no 'rule' about it.
Just per the Mom hosting it.
So you need to ask.
And then per your child... how he is at being dropped off, or you staying.
And bring a token something to the playdate. Be it snacks or some little thing. Not a 'gift' per say, but just a little something.
Usually, for me, the Moms bring some sort of snack. Bought or homemade. Its just courtesy.
or they ask if they can bring something.
AND this is the first play date at this Mom's house. You don't know her well. So are you comfortable with her level of supervision etc.?
Or just call her and ask!
I did that when my daughter was invited to a friend's house that had a pool. I simply ASKED the Mom, IF she will be there supervising the kids while they are swimming??? Because, my daughter is not a strong swimmer etc. I made it a part of pleasantry conversation.
She took no offense at my asking.
She would do the same thing.
I also ask the other Mom, if "lunch" will be served or only snacks? Then that way, I know. Then I can plan accordingly.
ALSO: with your son, you explain to him, what you expect etc. Does he know your phone number???? ALSO leave your phone number with the Host. Tell your son to call you ANYTIME if he needs to. Its okay etc.
And if he has allergies, you tell that to the Host.