Does everyone have to start residency at the beginning of the summer? Any chance of getting a psychiatry residency that will allow me to start at the end of the summer next year?

You can check on FREIDA, by program, to see if they offer alternate start dates.

You can also ask the programs when you interview. They may give you your vacation first, or just modify the timeframe.

Another option is to just keep your eyes open for spots that may become available when you are (but this is low-yield).