Specifically, I was thinking of the case where the R group is a carboxylic acid functional group or an amine group. Then there would not be a chiral carbon atom. Is there a restriction on what can be an R group that I am missing?
The alpha carbon is a chiral carbon atom, with the exception of glycine which has two indistinguishable hydrogen atoms on the alpha carbon. Therefore, all alpha amino acids but glycine can exist in either of two enantiomers, called L or D amino acids, which are mirror images of each other.
In general, there are a variety of amino acids that are not chiral (achiral). In physiological proteins, the amino acids are generally alpha-amino acids (one carbon between the amino and carboxylic acid groups), but there are also, beta-amino acids, gamma-amino acids, etc.
Just limiting ourselves to alpha-amino acids, other than glycine, the most interesting example of an achiral amino acid is amino malonic acid, which has been given its own three-letter abbreviation (Ama) along with the more-common alpha amino acids. Ama has being found in the hydrolysates of E. Coli and human proteins. See Aminomalonic acid: identification in Escherichia coli and atherosclerotic plaque PNAS vol. 81 page 722.
Also, the amino acids of protein orgin generally have an alpha H, but if this H is substituted the compound is still an alpha amino acid. If the two side chains on the alpha carbon are the same, the compound is achiral. For example alpha-methyl alanine. An example that occurs naturally in cranberries is 1-aminocyclopropane 1-carboxylic acid. The alpha amino acids with two side chains are sometimes referred to as alpha branched amino acids, and have been found in meteorites.
Another class of achiral alpha amino acids are those with a side chain double-bonded to the alpha carbon such as dehydroalanine.