I got an order for a fondant-covered cake but it’s supposed to be dairy-free...so buttercream is out of the question. 

What can I use to fill and frost the cake?

My son is vegan, and I am almost there too, so I have collected/developed a number of vegan recipes over the years.  If it must be vegan, you can't make IMBC or SMBC (eggs), but if it is just dairy you need to avoid, that is easier.  Sub vegan butter (such as earth balance) into your ABC.  Vegetable shortening, including high ratio,  is still fine.  There is also non dairy cream cheese available -  I much prefer Kite Hill over Tofutti (Whole foods).  Non dairy butter would probably actually work with SMBC come to think of it if eggs are OK but I haven't tried that.  If someone has, let us know!  If you need chocolate frosting, I have a great vegan, dairy free recipe I have tweaked over the years.  You can also make dark chocolate ganache with coconut milk, if a little coconut flavor is OK.  Semi sweet and bittersweet chocolates are usually vegan - read the label. (cocoa butter is not dairy). Don't  skimp on the flavorings and liqueurs.  If you choose to add butter extract, check that it is non dairy.  McCormick's butter extract has dairy products in it.  The spice house natural butter flavor is one that is vegan that I know of.  HTH.  

You didn't say what flavors you were considering, but you could also use fruit fillings - raspberry, strawberry, pineapple, etc for between the layers.