You can use WebDBGET to get data from KEGG.

You mentioned that you have gene names. You need to construct two queries.

1. For a given gene name (e.g. PGM1), use bfind to get the gene entries (e.g. hsa:5236, in human) in all species

http://www.genome.jp/dbget-bin/www_bfind_sub?dbkey=genes&keywords=PGM1

It produces a list of gene entries from several species. The gene entries for human and chimp are shown below

hsa:5236
PGM1, CDG1T, GSD14; phosphoglucomutase 1 (EC:5.4.2.2); K01835 phosphoglucomutase [EC:5.4.2.2]
ptr:456908
PGM1; phosphoglucomutase 1; K01835 phosphoglucomutase [EC:5.4.2.2]

2. For a list of gene entries (e.g. hsa:5236, ptr:456908), use bget to get both nucleotide and amino acid sequences in FASTA format.

http://www.genome.jp/dbget-bin/www_bget?-f+hsa:5236+ptr:456908

A relevant thread:

Is There Any Way To Retrieve Genes' Sequences In Fasta Format Using The Kegg Orthology Code?

You can use Ensembl BioMart to export the sequences in FASTA. Choose your Gene IDs (whether it's gene names or otherwise) as FILTERS and for ATTRIBUTES, choose Sequences and then select the option you want (e.g. cDNA sequence, coding sequence, protein, etc).