Standardize the log transformed data, if you are going to tranform, not the original dataset. The goal of standardizing (aka auto-scaling) is to have mean zero and variance 1 data input into PLSR, so as to force a y=0 intercept in model, and to have PLSR initially weigh all variables equally, respectively. Auto-scaling can introduce noise, however, so it is not always a good option. There are other types of scaling that are compromises between auto-scaling and just centering, such as pareto scaling, where you divide by the square root of standard deviation before centering. If you do not scale, larger magnitude variables will be weighed more heavily in PLSR than lower magnitude variables.
Log transformations are often used when your x variables are e.g. chemical concentrations, that are known to be lognormally distributed in the environment. But in other applications, (e.g. spectroscopy), log transformation may not make the most sense. There are other transformation options, such as the rank-based inverse normal transformation, Box-Cox, or Tukey that you could consider, that will almost absolutely force a normal result. What matters is that when you run PLSR, the relationship between the x-scores and y-scores are linear (t vs u plots). If you see a lot of curvature in these plots, you could probably obtain better results if you were to transform something.