My youngest comes nearly every day home with scribbles of white board markers on her shirt. I scrub them with the old fashioned Sunlight-block-soap, a good scrubber and piping hot water and if necessary repeat, before it goes in the washing machine. Now the carpet doesn’t go in the washing machine but with scrubbing with Sunlight block soap you came far

I also use hand sanitiser or alcohol wipes

Yes I’ve been using hand sanitiser too and it often works well

depends what the marker pen is on. If it is a hard surface then you can remove it with a small amount of cream cleaner, or failing that autosol, on a cotton wool ball, work in in, then another cotton wool ball to buff it all out.
If the surface is painted you have to be careful what you use as many of the things suggested here will stuff up the paint surface.
start with a baby wipe, then try something more substantial. Most of the markers for whiteboards are in theory not permanent, so at the very least you should be able to lessen it to a ghost mark.

Vinegar is supposed to be great at ridding stains

Hand sanitiser or rubbing alcohol usually works!

I’d try something like hand sanitiser or vodka, or some sugar soap.

Maybe sanitiser. I have been using sanitiser to remove ink of other things

Adding to my comment- hand sanitiser, pat to draw it out- another one that works (for permanent marker) is hair spray!

Hand sanitiser- liquid types work best, as long as high alcohol content.

Carpet can be tricky, depending on what it’s made of. I would google for answers first.

WD-40, it has so many uses, and it has worked for me in the past.

I would probably try the old fashioned Sunlight block soap

I use room freshener when it’s on a hard surface – yes, it really does work – but I don’t know if that would work on carpet.