Does anyone know how to back up an iPhone that has a completely black screen? It does vibrate when I plug it in.

If you have previously backed up the phone to your computer and opted to “trust” it the first time you should be able to connect it and back it up again, entirely from the computer.

Have you tried a forced restart?

• Press and quickly release VOLUME UP
• Press and quickly release VOLUME DOWN
• Hold the SIDE button until an Apple logo appears (15-30 seconds), ignoring "slide to power off" if it appears

The operative word here is QUICKLY; and try it a couple of times, the timing is a little tricky.

You cannot back it up. Your data will be lost when the phone is repaired.

If you back up to computer you can check the box to Encrypt this Backup to save your data. When you plug it into computer does it show up in Finder window?

How would you type in the passcode?

I humble bow to you! Although you might be able to approximate where the correct buttons would be if you overlaid a printed screenshot of that window. Maybe?