It's hard to be without a digestive pathway
The invention of the mouth wasn't necessarily about an operational method to gather food. It was a logical outcome from the very origin of animal life.
Essentially we are all based on tubes. The tube is an efficient way to evolve and grow, as essentially it is a simple genetic set of instructions that could change size and length over generations. Early genetic dead-ends, such as fractal structures, could not lead further than a plant-like fractal structure evolutionarily.
As nutrients pass through the tube (actively or passively) it is also absorbed through the edges of the tube to integrate within the rest of the structure, and we can grow ourselves.
We then evolved appendages (sexual reproduction organs, limbs, skeletal structures, lungs and so on) over time such that we are now very complex tubes.
In your case can an organism evolve without the origin of a simple digestive system? Can a fungus essentially grow organs and internal complex structures? The fact they haven't in several billion years of evolution likely means it is difficult for fungus to evolve beyond far from their current forms, much like early fractal organisms.
But we don't know that much really
However life is diverse, and much more complex than we realise. Animal life is but a very small slice of the amazing world of life. Often there is such variety we struggle even now to make sense of it, and are continually revising our understanding the more we discover.
From the above diagram there are many other types of organisms and they all have different efficient solutions to eating and multiplying, and not necessarily like an animal.