Can I convert a non-work visa into a work visa once I arrive in China?

I have applied to be an English teacher in China.
I'm suppose to be departing to China this August.
But my school forgot to send in one of the documents that I need for the Z Visa.
Either the "Invitation Letter of Duly Authorized Unit" or the “ Foreign Experts Work Permit” is missing according to an intermediary.

Can I apply for another type of visa and then convert this visa to the Z Visa once I arrive in China?
Or, can I apply for another visa once in China?