You really cannot do this without getting permission from the manufacturer. This would amount to an infringement called "reverse passing off." This is where you take one's product and relable as your own passing off to the public as if you are the source of that product when indeed you are not.
This is very easilly remedied, however. You can enter into a "white labeling" agreement with them or another source that allows you to relabel everything under your own brand. Many will even offer you the added service of labeling everything for you.
To keep it real, if we are talking about very generic made in China merch. it is very unlikely they would seek to enforce against this or frankly even care about it (although possible). But certainly if this is a producer of branded products I would not attempt it. Further, you also need to make sure that you do not misrepresent the country of origin.
If you intend to sell under your own brand I would most definately make sure to conduct the proper trademark due diligence (see link below) before making a significant investment in it. Once it clears then it can be filed with the USPTO so you can claim your own TM registration on it.
It might be helpful to discuss this in more detail with a lawyer in private. Most of us here, including myself, offer a free phone consult and I will link you to some general helpful info below.
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As noted by my colleague, even though your "wholesale supplier is not a popular brand" you may not lawfully remove the branding that it puts on its clothes [logos, hangtags, etc.] and then sell those clothes -- either unbranded or with your own branding. That constitutes, at least, reverse passing off. As also noted, you can buy unbranded clothes from a wholesaler and then affix [or have the wholesaler affix] your own branding. Some dog clothing is protected by a design patent [you can view some via the link below] so do not buy any counterfeits of these products or make any that infringe these patents.
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Totally forbidden. I have experience in relabeling litigation and it is brutal. It isn't only trademark infringement litigation but unfair trade practices in states like California. By the time you alter the clothing and insert a label you may as well design and manufacture your own line and avoid the $20 - 50,000 legal fees, which would be conservative here in Los Angeles
The above is general legal and business analysis. It is not "legal advice" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
My colleagues have answered you well. Since you want to establish your own brands, which is wise, I think you want to have the supplier provide you unlabeled merchandise. That is, let THEM remove labels. I disagree with having them affix the labels, as a general policy, when dealing with offshore suppliers, as you risk the "one out the front door [to you] and two out the back door" [to rip off artists, such as for sale on eBay] of branded product. The excess production will likely occur anyway, but at least not under your label if you do the labeling. If you have a decent labeling machine (and they are not very costly) you can likely reduce the number of fakes by doing your own labeling.
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