There's plenty of APIs that will let you do this if you want to use an API. You can probably do it with Blockchain's API but if that doesn't work I'm sure Coinbase et alii have an API that will work for you. If that fails, then you could always just use a lightweight client if you don't want to use Bitcoin Core, like Electrum.
APIs may be the way to go here. Grab unspent outputs you have access to from an api, sigh them offline, and then push the signed transactions through a pushtx-style api. Or, use bitcoinj, keep connected as an spv wallet, and push transactions as you see fit. This stops your dependency on an external centralized service, and gives you more flexibility.
