Virtually every AT/ATX standard supply still floating around can power an IBM PC 5150, with a little bit of work. I believe some people designed small kits, and even adapted the red toggle switch that can run off newer guts. I think you might be able to use a PicoPSU (ATX) even. So if you can't find one that is in the exact IBM PC form factor, you certainly can get a PSU. Although there are likely some IBM PC form factor PSUs still around.
Note, if you have an original 5150 A with 4116 RAM, you need the -5V rail too. Please tell us your exact model.
Also note 2, that the 5150 can take a very long time, depending on how much RAM is fitted, compared to modern systems, to POST. You'll want to wait at least 2 minutes to make sure.
I don't know what you mean by the "screen doesn't turn on". How are you powering the display? What is the display type? Are you trying to say that there is no signal to the screen, but you are providing power to the screen?
More information on the situation will be helpful. You'll want to narrow down and identify the root cause of your problem, and explain everything you find.
https://minuszerodegrees.net/5150_5160/MDC/mi … diag_config.htm
Yes, having another power supply on hand can help in the diagnosis. You can use an ATX to AT adapter with a common ATX supply you have around.