Firstly, it's the residuals from the model that need to have a multivariate normal distribution, no the observed values. Additionally, small deviations that some statistical significance test picks up with largish sample sizes may be irrelevant (and conversely such tests may miss really severe deviations in case of small sample sizes). So, ignore formal tests and look at how bad the deviation really is. If you then still have a problem, read on.
Secondly, transformations as you mentioned may often help (e.g. cell counts in the blood, viral load and similar variables often need a log-transformation).
Thirdly, if there is some distribution that describes your data well (whether it's for categorical, ordinal, count, continuous or whatever data), then a model with that distribution and a multivariate random effect (the simplest version is where you assume all times to be equally correlated and have just a single random effect per person - that's what a GLMM typically does, which covers many standard distributions like normal, binary, binomial, Poisson etc.) would be a logical option. A typical version with full flexibility of that for multivariate normal data is the MMRM model (mixed effect model for repeated measures), which allows an unstructured covariance matrix between times. Getting that flexibility for more complex situations with more unusual distributions may be less well supported by modeling packages (but e.g. things like Stan as usable via rstan, brms and so on allow extremely flexible model specification, it just is more time consuming to code up so many details oneself).
