How do I open up a small under-the-sink bathroom cabinet?

You can add a shelf inside if you wish. Or take the door off, insert a shelf and use baskets to keep things organized. I have a similar style bathroom vanity. I use some of these:

https://www.bedbathandbeyond.com/store/product/salt-trade-storage-baskets-and-shelves-collection/222248?categoryId=12100

I’ve used that space for a whole lot f pits and lids , writhing easy access With some muffins holders

attaching pic.

Since that looks like a standard plywood cabinet, check inside and see if you could use a jig saw or multi-tool to cut a larger opening to accomodate 2 doors. Then use blue painters tape to frame off the the cut area... look at this video to see how the base cabinet looks, you can always add some support wood to the inside of the cabinet out of sight.... https://www.youtube.com/watch?v=yMPcBpe6Mjs and here is how to hang the doors....

Hit up the Habitat for Humanity store for some used ones or see if they have some scrap plywood to cut two doors. Sand them, stain and poly, then add your hinges.

Putting a shelf in is easy.... just add two shelf supports inside the cabinet wall using a brad nailer from outside into the support, use a nail set to recess the nail head and use putting and paint/stain to hide.... make sure its level, and then measure inside for the length of the shelf to sit on top. I like to add a shelf that is shorter in depth than that of the cabinet so I can still reach into the back of the cabinet.... using 12" to 8" stock lumber.

I would add a shelf or two depending on the height of the cabinet. For easier access you might consider installing a wire basket on tracks so you can pull the basket out or forward for simplier access. We did this in the spare bathroom (which is tiny). Really did make it much easier to reach the back of the cabinet.

Sharon tells you the best way above. The easiest way (in case you aren't handy) is to remove the door and use two double tier lazy susans inside so you can rotate your stuff toward the front as needed. (Assuming you are wanting to store smaller items - this idea wouldn't work well for towels and larger items.)

I would cut the opening larger and add a sliding barn door over the opening and add shelves inside it....

I would put metal shelves, the kind you just put in place without any altering of the wooden structure, then add baskets.