Selling bags with someone else's logo or trademark on them is probably trademark infringement. While it may seem innocent enough, the trademark owners have the exclusive right to decide how their trademarks are used. So while bags without trademarks are fine to make and sell, ones with company logos are not.
The issue of repurposing or upcycling products or packaging containing trademarks into new products is a grey area of the law. In general, one can resell products (you see this all the time on eBay and on garage sales where used products are resold more-or-less in their original form with the trademarks intact). This is sometimes referred to as the "first-sale" or "exhaustion" doctrine. However, this doctrine may not apply where the product has been materially altered. For example, constructing a new camera from used disposable Kodak cameras, where the KODAK mark still appears on the lens, could be deemed trademark infringement if a consumer is likely to be confused into believing that KODAK is the source of this "new" camera. See Eastman Kodak Co. v. Photaz Imports, Ltd, 853 F. Supp. 667 (W.D.N.Y. 1993). The likelihood of confusion may be less if the appearance of the recycled trademarks on the "new" product is partial and incidental. If you intend to sell and market these products, you should consult with a trademark attorney about your proposed product in light of these issues (and which may include further legal research) before doing so.
This answer is for informational purposes only. It is not intended as specific legal advice regarding your question. The answer could be different if all of the facts were known. This answer does not establish an attorney-client relationship.
Both my colleagues have provided excellent responses. Generally a product that includes MANY TMs isn't infringing on any one of them because no customer could reasonably think that any one of them endorses or is affiliated with your company, but it does really depend on the exact use in question.
The best course of action is to consult your own IP lawyer before turning this into a business. You'll also want to check out whether the plastic bags are being discontinued by the retailers for conservation reasons.
Avvo doesn't pay us for these responses, and I'm not your lawyer just because I answer this question or respond to any follow-up comments. If you want to hire me, please contact me. Otherwise, please don't expect a further response. We need an actual written agreement to form an attorney-client relationship. I'm only licensed in CA and you shouldn't rely on this answer, since each state has different laws, each situation is fact specific, and it's impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue.
This sounds okay from a trademark standpoint, as it seems clear than no one can reasonably think these are endorsed by any one store if they have a bunch of partial logos from different stores. And no one store can legally object is there are many store logos appearing, as the many logos makes it hard for them to claim the consumer will think they alone sponsored this. Do NOT use all bags from one store if you want to be safe, use bags from many stores for each one of your "Bag Purse" purses and you should be legal. You will increase your risk if you use high-end brand bags with low end bags [e.g. GUCCI bags with DOLLAR GENERAL bags], but even then I am reasonably confident the high-end brand would think it counterproductive to publicize the issue by objecting. Another option to lower risk would be to put some recycling logo (get permission if not your original logo creation) prominently on the purse. That would also help you market the purse and perhaps give you something you might protect and that might generate value long term for you if you want to expand into other items. ["rePURpoSE purse"?] Consulting an IP lawyer might be a good investment if this seems a viable business for you. There may the potential for a patent if you have a unique and original way of doing this, and if so time is of the essence in securing patent protection due to recent changes in the patent law.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.