Can you sit shiva on Shabbat?
No, one does not sit shiva on Shabbos.
The reason for this is that Shabbos is a time that transcends the sadness and mourning of the shiva week. Therefore, one does not sit low, and do other acts of mourning on Shabbos, even in the shiva week.
The exception to the rule is marital relations, which a person must refrain from even on Shabbos.
For a summary of the relevant halachos see Kitzur Shulchan Aruch no. 219.