Does the rolling code encryption used by add-in boards for some of the Vertex models pass through an analog repeater?
Yes, but one caveat - keying a repeater takes time. If that time interrupts the synchronization, the short answer changes to no.
Your receiver has a squelch and CTCSS detect system which take a little bit to come up to speed. Some more elaborate repeater systems use a voting comparator which introduces more time into a keying sequence, and some repeaters use "in cabinet" repeat to work around a comparator or remote elements that might fail. Your radios also take time to detect a signal. Seeing past the jargon, parts of systems introduce delays. Your system will work if the repeater keys up and the radios decode BEFORE the synchronization burst is sent. If not, the radios will lose the first few syllables and will sync up much later than PTT. You may be able to program the delays in the radio to extend a "preamble" (that's what Motorola calls it in MDC-1200 terms, I'm not sure what Vertex would call it) time so everything is in position before the sync is sent.
Some systems could also have buggy issues, like weird audio frequency response that could affect decode.