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:
空手の修業は一生である。
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 [0] 【修業】
(名) スル
学問・技芸などをならい修めること。しゅうぎょう。「師について—する」〔同音語の「修行」は仏道に励むことを主にいうが、それに対して「修業」は学問・技芸などをならい身につけることをいう。なお「修行」は「修業」とほぼ同等の意で用いられる場合もある〕
しゅう ぎょう しうげふ [0] 【修業】
(名) スル
学問や技芸などを習って身につけること。しゅぎょう。
しゅ ぎょう —ぎやう [0] 【修行】
(名) スル
① 学問や技芸・武術などに励み、それをみがくこと。「—を積む」「武者—」
② 生理的欲求を禁じて精神および肉体を鍛錬することにより、精神の浄化や神的存在との合一を得ようとする宗教的行為。
③ 《仏》戒律を守ったり、悟りを開くために特定の宗教的行為を行なって、仏の教えを実践すること。仏道に励むこと。→修業(補説欄)