The leaves get these small light tan speckles and then get worse from there, turning yellow and crispy brown. It seems to affect mostly older leaves. The tomatoes are fine. They get west sun, could it be sun scorch? Or something nutritional or a disease?

It is growing in an earth box. I planted with a strip of organic granular fertilizer. Is it possible it has already used this up? Should I add a liquid fertilizer to the water reservoir?

Zone is 8b DFW TX

Research chlorosis, but with the amount of rain DFW is getting, you are probably looking at the reason. The green color of the plant is perfect, not sure what NPK you are using, but granular is good as it usually contains trace elements the tomatoes need, but that isn't an issue. West sun isn't bad yet, tho most plants prefer east and south sun with evening shade. I would simply remove those leaf branches that are going bad, ( sometimes water spots sunburn leaves) and be prepared to either put an umbrella over the plant for rain and sun protection. ( or shade cloth?)

I think it is spider mites. Now I see what looks like small grains of pepper under the leaves. Despite all the rain this weekend. They don't seem to move though but can be squished. How do I get rid of them without pesticides? I don't want to harm the beneficial bugs I see on there like spiders.

Insecticidal soap. There are recipes for making it cheaper at home than the ones you buy at Ace, or HDepot, etc.

I just tried some earlier that I had already bought, and the spray nozzle didn't work properly when trying to spray it upside down (to get the underside of the leaves). Very frustrating. I ended up running the leaves between my hands to coat them but I still couldn't get to them all. Does spraying just a strong stream of water work, like it does for aphids?

The soap kills them, aphids aren't killed by spraying water, and they will come back, if you bought that little sprayer the stores sell, yeah, it can't reach the liquad upside down. I just brush the leaves up and spray it on them. By the way, ants farm the aphids, treat them like little honey pots and tend to them.

There aren't any aphids that I can see. Seems to be spider mites and maybe thrips. I was just wondering if I could knock them off with the water because I can get it to spray under the leaves a lot better than the soap.

If you confirm you have spider mites, get a miticide or a pesticide that's also effective on mites(IE abamectine) and spray it. I am yet to see a household remedy that's not extracting nicotine from cigars (don't do it) that takes care of spider mites. They are really really tough.