Is there any way to upgrade a stadium without coins?
Nope, but I highly recommend you to try and switch clubs. It's much better to switch to a much bigger club. I went from a Prem Winner Middlesbrough team to a Mid-Table Lazio team because their facilities were 6/5/5/5 than Middlesbrough's 4/3/3/3. Once you are in a club with a Level 6 stadium, you don't need to worry about upgrading!
i wanted to win prem with tranmere before i move clubs.. you're right its hard to develop a stadium.. at tranmere i had everything else maxed but the stadium was an issue that was always going to hold us back.(i guess it is realistic). time to move on i think.