I know enough not to cancel my old card and apply for the new one because then I would lose my credit history from the old card.

You don't 'lose' the history. The history for a closed account will stay on your report for another 7 - 10 years. What you would lose is the boost to the average age of active accounts that this account provides, plus you would be adding a new account into that calculation. You will probably also end up with another credit inquiry on your history.

My question is, can I request that Chase switch me over to the Amazon card without losing my credit history?

You can ask, but you may not get. Sometimes accounts can be moved into a different reward structure without opening a new account, sometimes they can't. It depends on the rules Chase and Amazon have in place for the Amazon account. You might be better off seeing if there is another reward structure that you can be moved to, anyway, or if there is something that you can do (like open a Chase checking account) that will improve your current reward structure. Reward structures tied to a specific merchant are a lot less flexible in maxing out the rewards.

AJ