Can I legally teach in Thailand with a Non O visa?

To teach legally you will need a non-b + work permit. You will want to to get help with the paper work for the non-b. There are few different types of non-b. I am not that familiar but it's something like. Non-b retirement, non-b education, non-b marriage. The "Non-B" is kinda of dumb name.. It only means "Non Business " Visa. Most visa you get are non-b.

Take a look at the Thai Embassy website in your country, the list of requirements is listed on there.

If the school and you can hand over all the copies of documents needed it does take a few weeks.

The Non O can by given with several reasons, like marriage to a Thai, retired, volunteer, dependent, children etc, Not all of those allow you to work in combination with the work permit. If married to a Thai National it is no problem, maybe better option is to apply for the Non Immigrant B visa and get the work permit.

Your employer should apply for the work permit, on what basis did you get the Non O visa?