> For some reason, many sites, including amazon recommendations, fail to support 256 bit ciphers. I don't understand why.
I assume you mean AES-256 vs AES-128. If a site is preferring AES-128 ciphers (particularly in combination with other modern cipher suite choices (ECHDE, SHA2, etc)) that's actually a pretty good indication they know what they are doing.
AES-256 offers no practical advantage over AES-128. There are no known practical attacks against AES-128, and stretching the key to 256 bits just wastes CPU cycles.
In fact, most of the attacks out there that weaken AES to any meaningful degree are those that attack the key schedule used in the 192 and 256 bit versions.
tl;dr; Save the planet. Use AES-128.