"They raised their objections TO the company plan."
Is it possible to replace "to" with "at"? If not, please tell me why.

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If you want to say "at" instead of "to," you need to change the word "company" to "company's," and here is why.

The sentence "They raised their objections to the company plan," is very clear. The subject is "they," and the object is the plan that this company has drawn up, which is very clearly understood. But if you change "to" to the word "at," we get "They raised their objections at the company plan." This sentence is not as clear. The subject is still "they," but the object is no longer clear. It could be a plan to form a new company, etc. So if you change the word "company" to "company's," the 's at the end of the word shows that the plan belongs to the company, and the object of the sentence is clear again. So, put it all together and you get this:

"They raised their objections at the company's plan." Great sentence. You can also take out the word "at" and use "against" while still using the 's at the end of "company."

Hope this helps you :)