The brake and clutch pedals on my 1971 Land Rover Series III are a bit stiff and aren't returning fully to their rest positions. I suspect the pedal pivot is dry and in need of some lubrication. From what I’ve read, oil is recommended, but it’s not entirely clear which type of oil works best.

My thinking is that a thicker oil might be ideal—it would be less likely to drip straight through. I’ve also been advised against using multi-purpose grease, as it tends to dry out and become sticky over time. That seems reasonable, especially considering the hollow pivot shaft acts as a small oil reservoir, feeding lubricant to the bushings through a hole at the top. That’s how it seems to me it’s supposed to work.

Would SAE 90EP steering box oil be a good choice for this application?
Or has anyone had success using a semi-fluid steering box grease instead?

It's the opposite of what you think. The pivot is a basic bush and to get lube in there it will need to be thin, engine oil will do. Better would be RP90 penetrating oil if things are very stiff. Also check return springs are good.
Do not use WD40 it is not what it is for despite all the bumph you hear about it.
WD stands for Water Dispersant which is what it does.

I'm not sure if the Series 3 have them but my Series 2 has a threaded bolt in each pivot pin. From memory they are 1/4 UNF.
I fitted two 90° grease nipples and give them a shot of EP140 gear oil from time to time. I use a small grease gun like those found in the tool kits of vintage cars.

Looking at the manual, the trunnions and bushes are lubed with general purpose grease and the pedal,shaft that has the end plug in is removed and filled with sae 20 oil and reassembled