You've actually got a couple of issues: what to do with your luggage while in Kamakura, and how to manage your luggage while on the train. All of the local trains will have only relatively narrow overhead racks, and the Shinkansen to Kyoto will have slightly larger overhead racks. In short, hauling 29" suitcases from Shinjuku -> Kamakura -> Kyoto will be inconvenient, to say the least.

Best solution if at all possible is to use the baggage forwarding (Takuhaibin) service. I do not believe it's possible to arrange same-day delivery from Tokyo to Kyoto. You'll probably need to have your suitcases ready at some point in mid to late afternoon on the day before you leave, and the luggage will be waiting for you in Kyoto.
This means that you'd also need 1 or 2 smaller, overnight-type bags that you'd take with you.

Your hotel will arrange for the Takuhaibin service. They'll tell you when you need to have your bags ready. It would be a good idea, if at all possible, to have the name/address of your Kyoto lodging written in Japanese -- this should be available from either your Kyoto hotel's website or by emailing them.

I don't know if Kamakura has a station with baggage hold or lockers. If no one here knows, perhaps your hotel concierge or front desk would know or could telephone and find out. If there's no storage at Kamakura, you could take the train from Shinjuku to Yokohama, store your small luggage in lockers there, then head to Kamakura, stopping back at Yokohama en route to Kyoto.

http://www.jreast.co.jp/estation/stations/476.html

Above is a link to a map of the JR Kamakura station - the little key symbols indicate coin lockers - but don't know the size....

> If there are lockers, will my bags fit in those lockers?

If you do use lockers, when you consult the information about the size of the lockers (available on japan-guide.com), note that the stated size does NOT reflect the interior space allocated to the lock mechanism. I found out the hard way: I had a suitcase that I selected specifically to fit in a "medium" locker - and it would have, if I could have gotten it in, but I couldn't because there was a metal box inside the locker associated with the lock.

Also, if you do ever use a locker in a Japanese train station, pay VERY close attention to where the locker is! I once checked a bag, left the station, and only later realized that there were numerous entries and numerous banks of lockers. To my embarrassment, station staff had to help me find my locker. To my delight, they were able to do so!

I've heard that same-day takuhaibin services are increasingly common, but I'm not sure I would feel comfortable relying on it without a backup plan! I routinely used takuhaibin services, and always found it problem-free, but I must admit that I never counted on next day service - I always had an overnight bag with basic necessities.

You should be able to find lots of relevant info about takuhaibin, trains, lockers, etc. on japan-guide.com

Hope that helps!