On the generic question - interacting with Ethereum without using a client, it depends on the definition of "interacting". If it is limited to sending transactions only, then the short answer is yes, provided one:
- has the account's private key
- know the account's nonce
- has the called contract's ABI (not needed for simple payment transactions).
With the above, a transaction can be constructed byte by byte. One actual example is here. The constructed "raw" transaction can be sent over with tools such as etherscan's broadcaster (well, in theory even this can be rid of, but then one will have to implement - at least partially - the RLPx protocol) .
In fact, this is what makes it hard to censor a blockchain transaction: one can always construct the transaction off-line, and send the bytes through a post (or even voice) to friends, who can then help push to the network.
Although, the problem is the "one-way" sending can be hardly called "interacting". To make things go two ways, a receiver is needed, which will need to check the validity of incoming messages. This, in turn, requires fully implementing the protocol. Now we have just brought back "the client".
For the original question, calling a contract's method without using geth or parity, the same bytes construction methodology from the aforementioned post can be used, although the contract ABI becomes necessary. See this question for more details.