If you go to your database in SSMS and right-click, under tasks should be an option for "Import Data". You can try to use that. It's basically just a wizard that creates an SSIS package for you, which it can then either run for you automatically or which you can save and then alter as needed.
The big issue is how you need to transform the data. This goes into a lot of specifics which you don't include (and which are probably too numerous for you to include here anyway).
I'm certain that SSIS can handle whatever transformations you need to do to change it from the old format to the new. An alternative though would be to just import the tables into MS SQL as-is into staging tables, then use SQL code to transform the data into the 3NF tables. It's all a matter of what your most comfortable with. If you go the second route, then the import process that I mentioned above in SSMS could be used. It will even create the destination tables for you. Just be sure that you give them unique names, maybe prefixing them with "STG_" or something.
Davud mentioned linked servers. That's definitely another way that you can go (and got my upvote). Personally, I prefer to copy the tables over into MS SQL first since linked servers can sometimes have weirdness, especially when it comes to data types not mapping between different providers. Having the tables all in MS SQL will also probably be a bit faster and saves time if you have to rerun or correct portions of the data. As I said though, the linked server method would probably be fine too.