How should I go about sourcing for a packaging company or finding a company with competitive prices?

Shipping components are expensive and unless you have the volume to buy in quantity vs small bundles is when you'll be able to shop around for a better deal. For example, Uline offers packaging in bundles of 8 or 25 as their minimum, this is stock item for which they order several thousands units of to keep the cost down. For you it fits because you can buy in small amounts at a time. However, if you have a need for 1000 at least or more units then buying from Uline is really expensive compared to what you could buy them for. Uline is a major distributor with a 45-50% markup on their items (give or take) as they have large warehousing and infrastructures. If you are ready to move to the next level, first find the correct material name for each packaging component, and look for suppliers of that particular item. Search websites likehttp://www.thomasnet.com/ for suppliers. Or go through a packaging broker instead of a distributor, like Landsberg Orora, or Ernest Packaging, and they can do all the leg work for you and still save money.
Additionally, I know your drawing is not to scale, but there seems to be excess empty air space, if that is true then you can customize your packaging per exactly your needs, which will help to further bring the cost down. An engineer should take a look at your current situation because there is more at play than just shipping components - method of shipping, weight, weather condition, materials thickness etc all play a roll and can all contribute to cost saving while making the whole thing function better.
This question is common enough that I've written tutorials on it http://howtobuypackaging.com/how-to-buy-packaging-part-1-determining-your-packaging-needs/.

I always start by listing out what my packaging must do. Yours looks like thermal insulation, cooling, protecting the food in travel, expressing your brand, and perhaps presenting your food.

The ice feels expensive, is it dry ice? Would you consider having a return program, where you can reuse the ice.

But to answer your question directly, I prefer having my network do the vetting for me. Failing that, Google has been a friend.

Could you consider vacuum sealing the food to keep it fresh and then ship it in insulated envelopes?