meaning of "riding out on sorties in quest of adventure"

Solution 1:

This is not trivial prose to decipher.

sortie = a short, quick attack by a military force, such as a small group of soldiers or an aircraft, made against an enemy position

"A series of sorties was carried out at night by specially equipped aircraft.">

OR = "a short journey to somewhere you have not been before, often with a particular purpose"

Cambridge Dictionary

Hence: "They are always going out to search for adventure".

The second part of the question is "Whether they reap trouble or renown, they bear it for their sins".

reap = to obtain or receive something as a result of your own actions

Cambridge Dictionary

Hence: A sortie may result in trouble for them or it may result in fame. They reap {= receive} either trouble or fame.

bear = to accept, tolerate, or endure something, especially something unpleasant

Cambridge Dictionary

Therefore, whether they reap {= get from their sorties} trouble or fame, they bear {= accept} it as in penance for their sins {=sins they committed at other times}, or perhaps as a consequence of sins they committed during the sorties.