A quick fix would be to put a towel underneath the sheet.

Can you sew? I made all the sheets for my pack in plays using cotton knit and FOE (fold over elastic). If you wanted something waterproof, you could use PUL fabric, which is a knit on one side and laminate on the other. This is the same material used for cloth diapering. You can find FOE and PUL at walmart or online.

You could even make a pad to place under the sheet. I made waterproof change pads layering PUL, absorbent material and cotton knit on top. You could use a similar layering for a bedwetting pad.