I am completing a moderation analysis (model 1) in SPSS with the Hayes PROCESS macro, however I had previously applied a logarithmic transformation to my outcome and moderator variables for them to become normally distributed. Does it make sense to be using standardised variables like these in this analysis given that the output can't be interpreted in a 'real world' sense? If not, what alternatives are there to this? Some guidance would be much appreciated!
First, how to do things in SPSS is off topic here.
Second, OLS regression does not assume that the variables are normally distributed; it assumes that the errors are normally distributed (and this is only needed for some aspects of regression). But let's assume that the residuals (which we use because errors are unknown) are not normal.
Third, there is no statistical reason you can't include interactions (that is, look at moderation) with transformed variables. Whether this makes sense in your particular context is unclear, since you did not give us context, but, assuming that it does not, then you can use a form of regression that does not make assumptions about the residuals. Robust regression and quantile regression are possible methods.