Yes, it's possible, up to a certain length. But when you don't have enzymes doing the work for you, it's hard to get the exact sequence the way you want it after a certain length due to errors. Enzymes that do this type of work are highly conserved and highly specialized for their jobs. See this for chemical synthesis: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6270108/

Also worth noting is that protein structure is just as important as, if not more important than, protein sequence. Protein structure and folding is what conveys enzymatic activity and function. The tougher challenge in modern chemistry has been getting proteins to fold the way you want it to.