Is there a "definitive" database for Mobile Genetic Elements? This isn't quite my speciality, but I've been tasked with finding any evidence of Mobile Genetic Elements (like transposable elements, retrotransposons, bacteriophages, etc.) within some WGS samples. 

Is there a "definitive" database used with sequences? How "species-specific" would this be?

Furthermore, are there standard algorithms used to find these MGEs? I suspect we don't have a comprehensive database of all possible MGEs, but it may be possible to query against the sequences we do indeed have (or detect any repetitive elements within the genome).
Would advise doing Denovo identification method structure-based. Then use these public databases (  RepBase, Dfam ) to look for high similarity or homology exploration, this possibility if your species, the repeats not yet discovered, or deposited in the databases.

If the species well-investigated earlier and the mobile element recognised, I assume it will be a straightforward task using species-specific search via Repeatmasker tool with the -species flag to obtain the MobileElement annotation in the genome.

Let me know if you need any data from Repbase related to your species, i have an entree.