I have created a PSD, JPEG and PNG file for a school logo. Their t-shirt company wants a AI for the first time in years. Can I convert any of these files I have to an AI file?
You left out one really important point in your question--that you don't actually have Illustrator.
Based on that fact, the answer to your question is no. There is no way for you to convert a file into an Illustrator file without having the Illustrator program. There are no programs that I am aware of that can export to an .ai file format (maybe something like Corel Draw or AutoCAD, but even with those it is doubtful).
If the printer is asking for something in .ai format, they likely are anticipating needing to make some sort of edits. Depending on what type of edits, they might be able to work with a PSD file, which can be opened in Illustrator in such a way as to allow type layers (for example) to remain editable. I would make sure to speak with them ahead of time to get the exact dimensions they will need for their processing (including bleeds, etc.) then make sure the final file that you send them meets those specifications exactly, so they have minimal reasons to edit your artwork.
Assuming you have Photoshop, you can go into the channels palette and select the channel that give the greatest contrast between the logo and the background. Hopefully this is a high resolution image with just simple colors, preferably 2.
Hold down the Command Key (control on a PC) and click on the channel. This will convert the channel into a selection and you should see the dotted lines marching along your logo. At this point, go to the path panel and in the menu select "Make work path..." which will bring up an options menu. In the options menu you'll set the tolerance which is how closely to the selection your path will adhere - the lower the number, the more detail there will be in the path.
Once you click ok your selection will be replaced with a path and in the path pallette your path will appear as "work path". You can double click on that path if you want to officially name it, but that's not necessary. From here you can go to FILE>EXPORT>PATHS TO ILLUSTRATOR... which will give you an options panel to choose the path you want to export. Choose your path and select OK and save it to where you want it.
Some things to note. This will be a raw path and will have no stroke or fill applied to it, and if there are any cut out shapes, it won't have any compound paths in it. Whoever you give it to will have to do some additional work, but at least you can give them a vector version of your raster image. Best to work from a non-JPG file so there are no compression artifacts.
It depends on what the school logo looks like. If it's mostly text, then you could try what LeoNas describes. If it has more graphics, then you're better off manually reconstructing it with the pen tool in Illustrator.