Is it possible to use
sendrawtransaction without downloading the blockchain? Or will bitcoind reject the broadcast since it can't validate the inputs?
If the client does block it by default, is there any way to force it to send it?
The RPC call
sendrawtransaction tries to broadcast the raw transaction you give it by first adding it to its own mempool of transactions. If you don't have the UTXO database, then you won't be able to validate the inputs of the transaction, so your node will reject it from its mempool. So, essentially, no, there isn't a way with the default client.
You could modify your client to have a new RPC call that just sends the raw transactions without checking it. But then you run the risk of broadcasting an invalid transaction and having your peers increase your ban score.
Also see: Pushing tx to the network via AJAX, without installing bitcoin. That question lists a few APIs that have support for broadcasting a raw transaction, which may be helpful to you.