Yes, but if you are using a save that has already claimed the reward, it won't be available in the Anomaly even when in "expedition mode". At least I couldn't reclaim the Utopia Speeder.

This is the one drawback of the new ability to start an expedition from an existing Save. I used to do exactly this on a new expedition, as soon as I could access the Anomaly I'd claim the Utopia Speeder.

I have unclaimed rewards I could access, such as the Iron Vulture, but I don't want to "consume" them on an expedition. You can also "copy" your ship or multitool from your Primary save, but that requires nanites. I did clone my multitool though (S class staff with a massively upgraded everything). I'm going to try cloning my cloned multitool when I'm done with the expedition, to have even more ultimate upgrades.

Overall, I find taking advantage of the ability to load up supplies in the Anomaly for an expedition is well worth the loss of ability to claim expedition rewards. In either case, a nice ship near-ish the start of an expedition is pretty damn useful. And I'll probably clone this one back into my Primary save or just grab it again.