Are euphonium auditions relatively common? Certainly they are for trombone, including perhaps bass trombone, and obviously tuba. But is there a euphonium chair in most orchestras? Oh, wait, there probably is for brass bands or choirs. Might a euphonium player be expected to double on trombone or tuba?
Euphonium auditions happen all the time, but not for symphony orchestras. Auditions are usually held at conservatories and universities at which the student wants to study, and certainly for military bands. Often, those auditions will include excerpts from certain orchestral works, but those works are only part of the audition list. Most often, the audition list will contain works from the concert band or wind ensemble genre.
There are two Concerti for Euphonium and Symphony Orchestra I'm aware of -- one composed by William Pagán-Pérez and another by Robert Jager.
The euphonium is a relatively new instrument (first developed in the mid-1840s). Consequently, orchestral works written previous to that time didn't consider the instrument. Even today, the instrument does not comprise the standard orchestral low brass lineup -- two tenor trombones, one bass trombone, and tuba. (As we've seen, some composers add another trombone or even a second tuba, but this is not standard.) The high brass consists of three trumpets (often in C, though that is not a requirement) and four horns.
In professional orchestras, or even per-service orchestras, one of the trombone players is usually tasked to play any euphonium part that appears in the literature -- that player will be compensated for playing both trombone and euphonium. We saw that in this thread when Jay Friedman (principal trombone with the Chicago Symphony) played "Bydlo" on
Pictures at an Exhibition. The part is scored for tuba per Ravel's orchestration, but Friedman played the part on euphonium.
This Wiki article presents a
partial list of orchestral works in which the composer has written a euphonium or tenor tuba part (in German, the instrument is also called tenorhorn (not to be confused with the British tenor horn)):
Euphonium repertoire - Wikipedia
The good news is, repertoire for the euphonium is expanding and is much larger today than even 50 years ago!
The U.S. Navy used to require euphonium players to double on trombone, but no longer. Most reasonably competent and experienced players will play both instruments, but that isn't a requirement unless they state that in the audition packet.