There are drill guides and drill blocks available for making repeatable holes in stock. The usual suspects like Wolfcraft and Milescraft offer these, for example. Whether you use a block-style guide (either DIY or store-bought) or a fancy guide depends a lot on what tool you will be using. For example, a block is not going to help you if you are using a forstner bit.
(So, if you want to really make consistent 90-degree holes, read on. But, for your purpose of a climbing wall you don't really have to worry all that much. As long as you are close, the hardware used to attach holds will make up for it -- they are sloppy for a reason. No one at a climbing gym is using a jig. They are just eyeballing it and getting close enough. Even single-digit degrees won't matter at the end of the day. At most, you'd want a home-made block so you mostly get it right.)
Here's an example that I have some experience with:
Image from amazon.ca product page
I have an earlier model of this which I purchased to make a number of repeatable 90 degree holes in sheet stock with a 1-inch forstner bit. It works.... ok. The bearings in the chuck are not great, and the return action is a bit finicky. It is repeatable, and the holes are closer to 90 degree than what I could do by hand.
The Wolfcraft model has two springs, which I think would solve some of the finicky problems with the Milescraft one. But I don't really know.
There are also many DIY drill guides out there, ranging from simple tall blocks or fit-to-your-drill angles, all the way to clever versions of the store-bought ones. Some of the DIY options really depend on your drill having adjacent sides that form a 90-degree angle, though, which is why the fancy plunge jigs exist.
These links are sure to be dead in an internet minute, but search terms like "DIY drill guide block" will get you a lot of hits.
You can also use a 2-axis bubble level and attach it to the back of your drill, which is often close enough for government work.