If your question is using an mRNA transcript as a template for protein synthesis, well, you would need ribosomes, the transcript, tRNA and the amino acids to be added.
If you mean putting amino acids in a tube with the right solution to allow the dehydration reaction to add an amino acid to the previous one, with some sort of control to get a specific sequence, I'm sure it's been done, many times.
I think the process would be something like this:
1. Develop the necessary amino acids with a protecting group added to the carboxylic acid end.
2. Add the starting amino acid to the reaction vessel along with the solution. Add in the protected second amino acid.
3. Remove unbound amino acids. Remove the protecting group.
4. Add the next amino acid, and repeat.
Who was the first group to do this? No idea. I think the most common method for creating proteins is using recombinant DNA and placing it in an organism that develops the protein.