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While I can not find buses to Domrémy-la-Pucelle, (there may still be,) and taxis from the nearest stations may well be expensive, you can go the last bit by bicycle.
Google found me a bike rental in Neufchâteau. There might well be more, I am not familiar with the place.
That is a 12 km (7.5 mile) bike ride.
Other options may be a partial bus ride and walk for the rest, as mentioned in the question a taxi (and unless you plan to be there long, it can be advisable to ask the taxi to wait) or hiking all 12 km. Or hitch hiking, which comes with the normal warning to only do it when you would do it at home and know what you are doing.
And lastly, if it is not cycling weather, you can contact one of the businesses in Domrémy-la-Pucelle and ask them how to get there from 'here'. Places to stay often have that information on their websites but I did not find those websites yet.