Question

Can you still use a sewing machine if the feed dogs are not working

Answer 1

You could no longer do standard seaming, but you could do free motion embroidery. 
If you want to fix this there are a few common ways that they get broken and fixed from home:
thread could be jammed, just remove the needle plate and clean
hasn't been lubricated in a while, either take it in and have it done or do it yourself with a drop of sewing machine oil
or because they're dropped, in this case just look for a switch.
if it's not any of these, you may want to take it in and have it looked at :-)