Writing this out makes me realize how strange it is, but here it goes anyways. I'm going to be visiting another city where they have a store that sells vitamins and supplements that are inexpensive. I normally do this as the last thing before I return home, but the vitamins and supplements need to be kept cool. I take public transport (including buses and a ferry) and carry everything by hand (small suitcase with wheels and backpack). In the past, I've put ice in a bag with the containers, but I'm a bit afraid it would leak. It probably will be 5 or 6 hours between leaving the store and when I can get them into a fridge next. If relevant the vitamins and supplements are stored in the usual glass pill bottles. What's the best way to keep them chilled?
To keep your items cool over 5 to 6 hours you may need an insulating thermal bag designed to keep content at their temperature a bit longer than from a conventional bag. Such bags may be available at your grocery store, at a pharmacy, or even from a Pizza delivery service.
To additionally cool your items I would go for a gel ice pack rather than ice cubes because of the gel preventing leakage. These ice packs need pre-cooling in a freezer before use.
Apart from that I would check whether your vitamins and supplements are allowed to be transported at room temperature for a few hours. There are only very few substances that would suffer from such a short interruption of the cold chain.
Consider double-bagging your things for the trip.
My first impulse is to put the containers loosely in their own bag and zip/seal it shut. Put the sealed bag into a larger bag with the ice or a few frozen water-bottles. Separate frozen water bottles might be less liable to leak than loose ice cubes although they'll still get wet from condensation.
Put the whole thing into another waterproof bag.
An additional bag is a cheap precaution against a leak. Putting the supplements into their own container reduces the chance of a leak due to pressure of water pushing into the bag.
Loose ice cubes have the advantage that they can probably be purchased at the same time as the supplements and can be packed around the containers. Frozen water bottles have the advantage of ease of handling for the round-trip and reduced chance of leakage.
Try an experiment to see which alternative gives the best compromise while remaining cool enough to preserve your investment.
An insulating bag as mentioned by Takkat is a start, but icepacks would have to last the outward and return journey. If there is a supermarket nearby, you could buy some frozen peas or other veg and put that in bag with your vitamins and that would keep it cool at least for a few hours. Even if they thaw out completely frozen vegetables are perfectly safe to eat and refreeze.