I don't know exactly why 船越義珍 used 修業 since 修行 would also make sense. But since he did, I think the quote should only be considered correct the way you stated it:
Firstly, since you probably don't know, 修 is often pronounced しゅう (shū, long U), but in both 修業 and 修行, it may be pronounced しゅ (shu, short U), so there are three words, しゅぎょう (修業 or 修行) and しゅうぎょう (修業), which all have their own entries. (So for the record, at the bottom are the entries in the J-J dictionary 大辞林 Daijirin.)
修行 has a strong connotation with training/discipline in the "spiritual practice" (sādhāna) sense (see sense ③ below). 修業 on the other hand has a more "down-to-earth" connotation of studying/learning. But as its supplementary explanation says,
Still, sometimes 修行 is used with almost the same meaning as 修業.
Anyway, with the sentence at hand, it seems like
Karate is a lifelong pursuit.
is a good translation. (Using 修行, it would sound more like "Karate is a lifelong spiritual practice.")
I find it curious that in spite of Funakoshi's philosophical inclinations, he chose 修業. Maybe he indeed meant to focus on the down-to-earth day-to-day practical aspects of training. Or maybe he didn't and chose 修業 as an alternative spelling of 修行, to avoid the strong association with Buddhism. Or maybe he just didn't have any particular reason for choosing one over the other.
しゅ ぎょう —げふ  【修業】
しゅう ぎょう しうげふ  【修業】
しゅ ぎょう —ぎやう  【修行】