Use yellow mustard to make mayonnaise?
The mustard is used as a flavouring - either you like whatever flavour the alternative mustard yields, or you don't - and as an added emulsifier. The emulsifier effect can be achieved with pretty much any mustard, be it english, french, or in powder form (mind flavor interaction with vinegar in that case...). The mayonnaise might even work without any mustard if it isn't an eggless recipe, since eggs (lecithin!) act as an emulsifier too.