You might be OK regarding copyright, but you may a problem with the trademark law of "passing off."
The copyright rule of "first sale" means that the copyright owner's rights to control future transfers of ownership of a copyrighted item end with the first sale of the item (additional copies can't be made, however). Accordingly, you can sell or give away your purchased logos without permission.
However, if you market your product with "adorning" MLB stickers, you may be violating trademark laws; i.e., "passing off" your product as being endorsed by MLB. The prohibition against passing off is to prevent misrepresenting the source or sponsorship of the product to the public; e.g., that there is some sort of association between your business and MLB.
You should run your specific facts and plans by your intellectual property attorney, but at the very least, it might be better to sell your product as a stand-alone item and offer a "'free' MLB sticker with every purchase" (or something to that effect) with a clear disclaimer that MLB isn't affiliated with you or your product, and is not endorsing you or your product.
As my colleague correctly stated, this is a trademark issue, not a copyright issue, and you can't market your own new product with an official licensed MLB sticker sinec it'll give the appearance that YOUR PRODUCT is from the MLB. The essence of trademark infringement is whether the consumers are confused, and your proposed plan is to attract MLB consumers to your product and sell it to them.
Trademark rights are meant to designate the source of products and services and protect the buying public from imposters, who presumably provide a different quality of goods/services than the TM rightsholders. The idea is to protect the rights of consumers to know whose making the goods or services they're buying because of the perception (if not reality) of this difference in quality. So an MLB fans know that when they buy someting with the MLB logo on it, they're getting a ;product made or licensed by the MLB, and not by you or anyone else.
TM rights are applied for class by class -- books are in one class, jewelry is in another class, toys and sporting goods and Xmas tree ornaments are in another class, and clothes are in yet another class. If the MLB's trademark covers the class of products yours is on, or if the natural expansion of their TM rights, which are already very broad, extends to the class that your products are in, then as far as the MLB is concerned, you'd be treading on their TM rights and committing TM infringement. So if you wanted to put their stickers on your own toys, you're infringing the MLB's class 28 TM rights on toys and sporting goods.
See an IP lawyer to fully disclose your plans and get some specific advice.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Indeed, this sounds like a trademark problem. And when dealing with Major League Baseball, you don't want to even get close to that sort of "problem" -- they are very aggressive with their IP rights, both real and imagined.