If you want the changes to persist, then you have to assign them to the original variable e.g.
values$mydata <- values$mydata %>%
mutate(TotArtFillRate_Sem4 = ITRec_4/ITOrd_4,
TotArtFillRate_Sem3 = ITRec_3/ITOrd_3,
TotArtFillRate_Sem2 = ITRec_2/ITOrd_2)
And BTW there is no need to call mutate() so many times, you can do it with just one.
If you need more specific help, please make a minimal REPRoducible EXample (reprex) of your app, here is a nice guie about making a reprex for a shiny app.