While I can't find any indication of how they do it, they definitely do excrete waste.
In book 14, the animorphs discover an Andalite toilet.
<So? So what is it?> Marco demanded.
<lt's a disposable module of a type used in the old days on the first generation of Andalite Dome ships. When the modules were used up, they were jettisoned into space. They were supposed to be aimed toward a star, so they'd be burned up without a trace. This one must have drifted through space, eventually being caught by Earth's gravity.>
<So it's a space engine?>
<lt's a weapon?>
<No, of course not. It's . . . well, this is a bit embarrassing. It's an Andalite Dome ship's modular waste disposal system.>
For about a full minute, no one said anything. Then Marco spoke.
<You're telling me the Most Secret Place On Earth, the fabled Zone Ninety-one, the Holy Grail of conspiracy nuts, is hiding the secret of an Andalite toilet?>
<Only a very primitive model,> Ax said condescendingly. <Since those
days there have been huge technological improvements.>
Animorphs #14: The Unknown - Chapter 20
The toilet itself isn't that well described and not much details can be determined about its methods of operation.
It was about eight feet across. The shape was like a cube with the corners rounded off. The entire surface was covered with tubing and painted symbols.
At one end was an opening, large enough for a person to walk inside. I could just barely get a glimpse of the inside. It was smooth, a lovely green in color, with soft lighting. There was some sort of instrumentation on one wall.
Animorphs #14: The Unknown - Chapter 19
As far as I know, the closest that the books ever come to discussing the Andalite excretory system is in book 49.
"Amazing." Cassie's mom reached out and ran her hand across the blue fur on Ax's rump.
"Mom!" Cassie snatched her mother's hand away. "Would you play with Jake's butt?"
"Of course not!"
"Then quit playing with Ax's!"
Animorphs #49: The Diversion - Chapter 15