I have had a similar situation as you. I have two American green tree frogs (I think it's a male and a female) which I believe have tried to mate. However, without a rain chamber, I don't think they will lay eggs in the water bowl. I've seen my frogs in the "amplexus" position (where the male gets on top of the female), but so far, in my water bowl, there have never been eggs. Also, there probably wouldn't be enough oxygen in the the water in the bowl. Hope this helps!

Sanimal