I have many variables and 4 groups. I would like to run an ANOVA for feature selection before multiclass ROC analyses. Can I do this? Does it make sense? Is it statistically correct?
No. p-value selection doesn't work. I would recommend putting your ANOVA model in regression form and then use adaptive lasso.See: https://www.bing.com/search?FORM=U528DF&PC=U528&q=ANOVA+as+Regression and
Mátyás Bukva ANOVA can be used for feature selection but know you have other options available. David Eugene Booth has already pointed out an embedded method, LASSO. Here is a non-technical introduction to the more common techniques: https://towardsdatascience.com/intro-to-feature-selection-methods-for-data-science-4cae2178a00a.
One thing I missed in your question, what classification technique are you planning to use after feature selection and multiclass ROC diagnostics?