Can vinyl alcohol react as a dienophile in Diels-Alder reactions?
Firstly, your Diels-Alder products are missing the double bond in the ring (formed between what was formerly C-2 and C-3 of the diene). You could always simply hydrogenate it down to the alkane, but I feel like this would present some selectivity issues in your professor's suggested synthetic route, especially since a terminal double bond is more likely to be hydrogenated than an internal double bond (sterics), and an internal double bond more prone to ozonolysis than a terminal double bond (more electron-rich).

Anyway, ron pretty much said what there is to say about "vinyl alcohol". It doesn't exist in any appreciable quantity, since the keto form acetaldehyde is much more stable (see another of ron's answers here):



Apart from what ron suggested (vinyl acetate), another alternative is to use 2-chloroacrylonitrile as the dienophile. The cyano group is electron-withdrawing, so the the usual requirement for an electron-poor dienophile is met. Treatment with hydroxide will unmask the ketone functionality, see for example the second and third steps in Corey's prostaglandin synthesis. (For a possible mechanism, see Mechanism for basic hydrolysis of α-chloronitrile to ketone?.) You can then hydrogenate the C=C bond chemoselectively to get the desired product, cyclohexanone.



2-Chloroacrylonitrile is what is called a "masked ketene". Ketenes don't undergo [4+2] cycloadditions such as the Diels-Alder reaction because they have a much greater propensity to react in a [2+2] fashion (even cyclopentadiene, an excellent diene in the Diels-Alder, reacts with ketenes via a [2+2]). So, these reagents are essentially surrogate ketenes: with one additional step (hydrolysis in this case), you can access what is formally a ketene+diene Diels-Alder adduct. There's a nice review of masked ketene reagents here.