What markers are like sharpies without the fumes?
If I was looking for such markers in my area I would go to the shops where they sell toys and craft items for kids, specially the kind of shop where they sell the cheaper items.

They are not always available but often one or the other shop has markers with wider barrels and chunky tips.
Look in the toy section rather than in the 'grown up's' art section.

I would go for toy or kiddies supplies as you get many more markers for the same money and I do not think a 5 year old child minds if it is a lower grade, as long as he gets replacements while he still likes the activity.