He can certainly commission a sword - the price would be either finding his own metal for the blade, or about half the cunes a blacksmith in the city would charge, unless he can offer something else in trade.

As far as an apprenticeship goes... Lmao this is where things get awkward, I'm so sorry. Hiccup actually had pretty close CR with the previous Keith (they built Wilders' Edge where the forge is together and were roommates for months), so being around a Keith that doesn't remember him would be... awk. He can probably convince Hiccup eventually but it would definitely take some actual threading to get there.