There is a nicely written and highly-cited chapter by Finney & DiStefano (2008) that speaks to your questions (you can view most of it on Google Books). In summary, multivariate normality is typically evaluated using univariate skewness and kurtosis, and multivariate kurtosis--values less than 2, 7 and 3, respectively, are generally considered acceptable, though as of their writing, no simulation work had thoroughly vetted these cutoffs.
If your variables do not meet those criteria, could you still use ML estimation? Sure, and your parameters estimates (factor loadings, factor variances and covariances, etc.,) would be pretty accurate. Your standard errors and $\chi^2$ test of model fit (and therefore your other typical indexes of model fit), however, would be biased; the greater the departure from multivariate normality, the greater the amount of bias you could expect.
In most cases, and as Finney & DiStefano's (2008) review suggests, the most straightforward way to handle to non-normality is to use a robust ML estimator, that corrects for non-normality-induced bias in the standard errors, and produces a Satorra-Bentler (S-B) $\chi^2$ (and associated model fit indexes) that more accurately captures the appropriate amount of misfit in your model than the standard $\chi^2$ test of perfect fit (Satorra & Bentler, 2010).
lavaanhas a few robust ML estimators, though only the
MLM estimator produces the S-B $\chi^2$. I'm not familiar with simulation work comparing the S-B $\chi^2$ to other corrections like the Yuan-Bentler (Y-B) $\chi^2$ produced by the
MLR estimator, or their technical differences from one another. However, I have used both
MLR in other SEM software (e.g., Mplus) and they usually produce very similar results. You might also want to consider
MLM if you have some missing data to deal with (
MLM is for complete cases only), and then read up on how the Y-B $\chi^2$ is different from the S-B $\chi^2$.
Finney, S. J., & DiStefano, C. (2008). Non-normal and categorical data in structural equation modeling. In G. R. Hancock & R. D. Mueller (Eds.), Structural Equation Modeling: A Second Course (pp. 269-314). Information Age Publishing.
Satorra, A., & Bentler, P.M. (2010). Ensuring positiveness of the scaled difference chi-square test statistic. Psychometrika, 75, 243-248.