得る is usually read as える. This is the "modern" verb-form of 得（う） (shimo-nidan).
One thing to note is that for MZK and RTK, either conjugation is え, so you will never-ever read 得ない as "unai" etc, and because you never really see classical SSK in regular text (since SSK and RTK are read the same in modern Japanese) you'll probably never have to read it as 得(う). The only time you read it as うる is if it's a sentence ending or directly modifying something. But again, the うる reading itself is pretty rare. The only thing I tend to see is an occasional あり得る(ありうる), but you can read that as (ありえる) anyway.
For that particular sentence: 彼はとてもよく訓練された役者だけが本当に成功し得ると思っている
^I would guess 'しえる', just because える is more common. し得る on my IME comes up as a candidate for both しえる and しうる.
MZK=未然形=Mizenkei (ex: ない attaches)
RYK=連用形=Renyoukei (ex: て/た/ます attach)
SSK=終止形=Shuushikei (ex: end of sentence when NOT a question)
RTK=連体形=Rentaikei (ex: declarative, bound ending, question ending)
IZK=已然形=Izenkei (ex: ば (conditional in modern/"when"-only in classical))
MRK=命令形=Meireikei (ex: imperative "commands")