I had a similar task after a sequencing run. I mapped back the reads to the contig and where the coverage promptly elevated (in my case two-fold of the mean coverage) there was a repeat. It is a similar approach as aligning short sequences. Self dot plot can work good as well.
It is very general answer and you would need to search for exact parameters. If you align DNA sequence with the same DNA sequence and set up "loose" parameters (align for short sequences, like up to 10-20 bp, low homology (60-70%)), you should be able to pick up repeats within the same sequence.
I don't know which online program is the best, but short Google search shows NCBI BLAST and a list of align tools.
Quick way to check if your sequence may have repeats is to build online restriction map. Some repeats have the same restriction site in the same position, and if these repeats make tandem, your restriction map would show, for example, 5 sites for the same enzyme located 1 kb from each other. Better to use "frequent cutters" (4-6 bp recognition sites) and make sure you cover both AT- and GC-rich restriction enzymes.
If you don't see this pattern, it does not mean that there are no repeats.