Is this sentence from Fox in Socks grammatical?

Dr. Seuss was usually pretty good at grammar. Yet his children's classic Fox in Socks includes this line:

Who sews whose socks?

Sue sews Sue's socks.

Who sees who sew whose new socks, sir?

You see Sue sew Sue's new socks, sir.

The third line doesn't look right. Shouldn't it be sews?

(and that second who on the third line looks like it should be whom, but perhaps Dr. Seuss takes a more permissive approach, particularly as he has to make the words flow.)

Well, you can't say "see whom sews": that would be ungrammatical, because "sews" is a finite verb which would have to have a subject, but the form "whom" is usually* used only when the pronoun is not the subject of a finite clause. For comparison, we can say either "I see him sew" or "I see [that] he sews", but we can't say "*I see him sews". You'd have to choose between "sees who sews", with an embedded clause "who sews..." that has "who" as its subject, and "sees whom sew", where "whom" is the object of "sees" and "sew" is an infinitive.

In this case, the form of the answer ("You see Sue sew Sue's new socks, sir") tells us that the intended meaning of the question is "Who sees who[m] sew whose new socks, sir?" The use of the word "who" instead of "whom" in contexts like these has been common for centuries, so I would not consider it an error.

The sentence contains multiple interrogatives, which is rare but not impossible. That may have contributed to your confusion. In a sentence with a single interrogative, that word or phrase is usually moved to the front of the sentence, and the subject and verb are inverted with do-support if appropriate: for example, a sentence like "He saw Sally" corresponds to a question like "Who(m) did he see", and sentence like "He saw Sally do something" corresponds to a question like "What did he see Sally do?". However, when there are multiple interrogatives in a sentence, the later ones stay in place, as in the question "Who saw who(m)?", or "Who(m) did he see do what?"

The verb "see" can be used with an infinitive after the direct object describing what the object is doing; e.g. "I saw her do it." I think it's not used very much in the present tense, but the Dr. Seuss sentence is contrived and unnatural in various other ways. The construction is nevertheless grammatical.

The "Who sees who sews whose new socks, sir?" construction would also have been grammatical, I think (with a slightly different meaning: this implies that someone sees that someone else sews, rather than than someone sees someone else sew), but Seuss may have avoided it because interrogatives in embedded clauses like this are often not interpreted as requests for information. For example, a question like "Who knows who sews new socks?" might receive an answer like "I know who sews new socks," which isn't the right format. The occurence of the other interrogative pronoun "whose" later in Seuss's sentence might disambiguate it, but the "Who knows who sews..." start still might cause the reader to go down a "garden path" of misinterpretation. In fact, looking at "Who sees who sews whose new socks, sir?", I find it very hard to interpret it with the meaning Seuss intended for his sentence.

*almost always, really; there are only a few exceptions of disputed acceptability (which I won't go into in this answer, but which you can see described on the following page: The use of nominative "whom").