Given a writing speed of one logical symbol per planck time, would the amount of time it takes to write Rayo's Number exceed Rayo's Number?

No. The amount of time it takes to write Rayo's number is 1, in units of Rayotime.

Write it how? Seeing as Rayo's number is defined as the smallest number bigger than what can be written in first order set theory with a googol symbols, you could probably write down the number with, say, googol+1 symbols. Or let's just say 2*googol symbols, that should certainly be enough. It would not take Rayo's number amount of seconds to write down 2 * googol symbols.

TL;DR: No.