Have a go with this. It works for me on the limited example you provided, but it might over match with other lines. It only replaces the first one on each line; if there are multiple sets that need to be replaced it will need at least a small tweak if not more effort.
sed -r 's/<a href="([^["]*)"/\1&/' input-file >new-file
If you are using sed on a BSD machine, or using the sed which is a part of the AT&T AST tools, then use -E instead of -r.