In D&D 5E, is there a way besides the Guild Artisan Background (PHB 132-133) to learn professional skills? i.e. blacksmithing, knowledge, gem cutting, etc.?
Practice, practice, practice....
You seem to be describing Downtime Activities, found on PHB p.187. One class of downtime activity is "Training," whereby you can gain proficiency with a tool, like blacksmith's or artisan's tools.
The examples given in "Training" are pretty thin. You'll definitely want to/need to work with your GM on this one.
As an aside, in one of my groups we rule that downtime accrues as time passes in real life. (In addition to as time passes in-game.) That way when it's hard to get synchronous games in, at least we're amassing currency for the asynchronous game =)