Is it Travis CI necessary to develop an R package hosted in GitHub and the development a website with pkgdown?
It is not. In fact we suggest that you do not use Travis CI, but use GitHub Actions instead: R Packages (2e) - 21 Software development practices
OTOH, GitHub Actions are not necessary, either. Many people develop R packages without a CI.
Thank you Gabor.
Can I delete the .travis_yml in my packages?
If you don't want to use Travis, then sure.