I'd like to add onto samhana's answer.
Background about the term and reading
The ことわざ reading for 事業 is valid, but it is also ancient. This reading is not used in modern Japanese.
- The ancient ことわざ reading is attested in a text from 757, as indicated in Shogakukan's Kokugo Dai Jiten entry here.
(After a site redesign a while back, Kotobank's formatting makes it really hard to read the entries. The top paragraph under the bold heading, the first 〘名〙 section, is for the ことわざ reading.)
- This reading is a simple compound of 事【こと】 "abstract thing, fact" + 業【わざ】 "intentional action": doing something intentionally and with purpose. The meaning is thus roughly the same as modern 事業【じぎょう】.
- This is cognate with 諺【ことわざ】 from 言【こと】 "word" + 技【わざ】 "technique, skill", but it uses different senses of the underlying roots.
- Considering that multiple dictionaries still list this reading (which usually doesn't happen for Old Japanese-only terms), I suspect that 事業【ことわざ】 may be used some in Classical Japanese as well. In fact, looking at it now, I see that the entry in my local copy of Daijirin includes a quote from the 養生訓【ようじょうくん】, a text from 1712.
- At the same time, I also don't see any pitch accent information for this reading, which usually indicates that the dictionary compilers don't consider this to be used in modern mainstream Japanese (a.k.a. 標準語【ひょうじゅんご】).
Back to your questions:
When should the on'yomi and kun'yomi of 事業 be used?
In modern writing and speech, only use the on'yomi of じぎょう.
If you're writing something deliberately archaic, using Classical Japanese, you might be able to use the ことわざ reading. However, expect that modern readers will need ruby ([振り仮名]【ふりがな】) to understand that you intend this reading. Also, make sure you do a lot of research into how this term was used to make sure you understand it well enough to use it. (I certainly wouldn't be comfortable myself trying to use this.)
Moreover, shouldn't ことわざ mean 'proverb'? Does that mean ことわざ is polysemous?
Strictly speaking, ことわざ as a spoken word is indeed polysemous (at least, in older Japanese; see also the Japanese Wiktionary entry for ことわざ for even more homophones of different meanings, a.k.a. 同音異義語【どうおんいぎご】) -- but the kanji spelling 事業 is less ambiguous, and would never mean 諺.