Just need an answer to my question. What is (block 15 of W-2) Ohio's Employer's State ID number?

It is just to the right of the OH.  It might be in a format like 52-xxxxxxx or something similar.

If it's blank, and there's an amount in box 17, you will have to contact your employer to get that. Or you could try entering the federal EIN there. If boxes 16 & 17 are blank. you don't need the employer's state ID

The Employer's State ID  for box 15 is OH (the postal abbreviation). It is not a number, The Employer's State ID  # goes in the un-numbered box between boxes 15 & 16.