Is there an equivalent for the name 'Wayne' in Polish?

NT, all in subject

As far as I know, there isn't.

Yes, there is, even two.
If way means droga - Drogowski
If way means sposób - Sposobowicz.
Simple.

From what I know, 'Wayne' means 'wagon builder or driver', as in the old style covered wagon with horses. I'm pretty sure 'Wayne' doesn't mean 'way' as in 'path' or 'direction'.

Woźnica not that common.

Or Woźnicki.

Wozniak and lots of variants on that

Wożniak is quite common.

I like Woźnica, but if I use that will I get a lot of strange looks and have to spell it all the time?

Yes, people will consider you a horse cart driver. Similar to this guy:

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Hah! Wożniak it is then