It is possible to reconstruct aminoacids in a lab, but not possible to make twenty aminoacids and then to connect and fold them into proteins. For that one needs instructions as from the DNA in the nucleus.
Yes, they can be synthesized in a lab. The emphasis here on "synthesized."
syn•the•size \-ˌsīz\ verb
-sized; -siz•ing verb transitive 1830
1: to combine or produce by synthesis
2: to make a synthesis of
Inc Merriam-Webster, Merriam-Webster’s Collegiate Dictionary. (Springfield, MA: Merriam-Webster, Inc., 2003).