Yes absolutely! If you're really interested you can look into a very famous chemistry professor named Samuel Danishefsky - he carried out the total synthesis of a fully glycosylated Erythropoietin protein. This took probably ~200-400 Phd years (20 people ~ 10- 20 years) to fully complete as there are MANY challenges that come along the way.

Fundamentally, its much much easier to do semi-synthesis or coupling of large chunks of proteins that have been synthesized by cells in cell culture. If you're interested in semi-synthesis you can look at famous chemical biologists like Tom Muir.

The VAST majority of times, when researchers/scientists want a protein, they simply ask bateria/yeast to express it by adding in the DNA that encodes for that protein. The cells natural protein producing machinery produces it, the cells are killed, and that protein is extracted and purified. This is FAR cheaper and more reliable than other methods to date. IF you want to make an unnatural protein - you can simply just change the DNA sequence that you're transfecting the bacteria with.

SYNTHETICALLY we can make proteins from total scratch. Its a TON of work, but has been done before (mostly for the purpose of understanding and optimizing really hard chemistry - not for large scale production).

1. Coupling amino acids in LARGE iterations has poor yields. - if you make a protein with 500 amino acids, and want to couple them step by step, that would be (excluding intermediate steps) 1000 reactions. EVEN at a 99.5% efficiency you're going to have huge problems (0.995^500) = an 8.1% yield. What is EVEN worse is the remaining 91.9% of the material is VERY similar to your product and you will have issues separating it (unless you tag it and use special chromatography).
2. Chemical reactions become HARDER (substantially) as molecules get larger. Fundamentally to reaction kinetics is the "number of productive collisions". If two objects (molecules) collide and they're small a huge fraction of their surface (say 10%) is reactive. If these molecules are much larger the portion of their surface that is reactive decreases with the surface area of the molecule making is much lower (say 0.1%). This slow down in rate makes some reactions on a large scale simply too slow to function.
3. Mammalian Proteins are Glycosylated - this process is inherently heterogeneous. Most of proteins that are produced in humans and other mammals are covered with glycans (sugars) of varying length (1-20 monomer sugars). This post-transnational labeling of proteins is not consisten (same animal, same protein, may exist with a distribution of types/locations of sugars). Sugar chemistry is FAR more difficult that protein synthesis and this proposes a MASSIVE challenge in ensuring you get the "correct/natural" protein that is in your body. The exact source of the protein (yeast, human tissue, bacteria) will produce the "same protein (amino acid sequence)", but variants of their attached glycans.

NOTE* Anything with 50 amino acids or less (very small proteins or peptides) CAN be produced on scale chemically quite easily with a process known as Solid-Phase Peptide synthesis. For large scale, industries actually do liquid phase peptide synthesis as its higher yielding (but very labor/time intensive for a single person).