Where should I start looking for an online gene library?
Databases
There are too many databases to list here, but Wikipedia has a decent list of the genome databases. For example, there are databases dedicated to genes of individual species, like Wormbase, SGDB and countless others. Alternatively, as others mention a great place to kick off your search is the NCBI.
Finding your nucleotide sequence in databases
It sounds like you already have your sequence. If you want to see if your sequence has been found elsewhere or if there are any similar sequences, you can use BLAST from NCBI. Although it is not strictly a database in itself, it looks to align your sequence to similar sequences in external NCBI databases and quantifies the sequence similarity. It will reveal the species for you to go to a more appropriate database along with a lot of other information about the gene.
There are plenty of parameters to alter to suit your specific needs, but generally, the default settings are a great place to start analysing your gene. You'll need to run a nucleotide test since you want to find the DNA sequence. The results might be difficult to interpret.