My 5-year-old's gotten really into drawing all kinds of things with sharpies, but the smell is pretty awful (and mildly unhealthy). What marker has the look-and-feel without the fumes. The super-permanence of sharpies is not necessary. In fact, not bleeding through typing paper would be a plus!
You can get acrylic based markers of high quality, which are nontoxic. My favorite ones are Posca.
I personally believe that children should always have access to the highest quality art supplies, because cheap tools invariably lead to bad experiences. Bad experience turns people (and children are people) off from things that could otherwise be enjoyable and exciting.
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.
I liked the fat Crayola markers with the single sharp conical tip for my art kids at the YMCA when I taught there. I tried cheaper brands, but in the long run the Crayola ones lasted longer, worked better and ended up costing less. Be sure to teach your kids to put lids back on right away, and to press LIGHTLY and they should last a long time. If they do get dried out before they get emptied of ink, you can sometimes revive them by dipping the tip in hot tap water for a few seconds.
I believe you can get Crayola with tips on both ends. I've never found double tips to be a valuable feature with my own higher quality Copic markers. I never use the chisel tip. The shaped tips sometimes on offer (like stamped stars, forked tips, etc.) are just kind of a gimmick in my opinion.
For my classes I also bought thinner Crayola markers for the older kids. These offer a smaller tip and are good for greater detail, but if I were buying for my own use, I'd get the fat ones (or more likely some of both). They're not archival, or I WOULD use them. They're very nice.