Most of the diy aquarium stand plans I have found use a frame. Since
this style is frame less, I want to make sure it won't fall apart on
me. Will screws & wood glue be sufficient to hold it together.

Yes.

Would
you recommend the use of biscuits instead?

No - this is a topic of some debate here, but single biscuits probably
don't add much strength to a joint, although they are helpful for
alignment if you need help with that. If you wanted to strengthen the
design past the butt-joint+glue+screw plan, you could rout dados or
rabbets to increase the glue surface. If I were making something like
that, though, I'd feel perfectly comfortable just using glue+screws.
My only concern would be visibility of the ply edges, unless you're
veneering after construction.
Good luck,
Andy