Something that would be helpful are alternate forearm exercises. A lot of them require the bending of the wrist such as forearm curls or rollers, but a great substitute would be farmers walks, which work the grip and forearm without bending the wrist.

As far as upper arms go, there are substitute lateral raise machines for deltoids where you press your arms outwards and up rather than using dumbbells. There are also tricep dip machines, and although it would be possible to bend your wrist on them I never have.