Is galvinoxyl antioxidant assay possible using NMR spectroscopy?
Galvinoxyl is a free radical, which means it has an unpaired electron that can be detected with EPR. Antioxidants reduce the radical, which is then not detectable in the EPR anymore as it doesn't have an unpaired electron. What you need the EPR for in the assay is to determine how much of the galvinoxyl is still a radical.

Paramagnetic molecules like galvinoxyl are problematic for NMR, they bleach signals in close proximity to the radical and cause line-broadening. You still might be able to quantify the amount of oxidized and reduced galvinoxyl, but I'm not sure how reliable this would be. The oxidized galvinoxyl would probably be not visible in the NMR, while the reduced version would give a signal. You'd have to measure a few spectra to see if you can determine the percentage of reduced galvinoxyl from the NMR spectra. 

EPR is the far easier method here, while I don't think doing this with NMR is impossible, there are quite a few ways it could be inaccurate. I'd strongly recommend to either use an EPR machine somewhere else, or use an entirely different assay that doesn't require such specialized equipment. I have no experience with antioxidation assays, but I'd guess that some exist that don't require EPR.