Is it possible to determine the symmetric encryption method used by output size? I'm attempting to identify the method of encryption for a black-box symmetric encryptor that produces blocks of output that are 4 bytes in length (e.g. small inputs fit in 16 bytes, then 20 bytes and 24 bytes as more input characters are added).

It's symmetric encryption and the value is always the same for the same input text. Is it possible to determine which method of encryption is used? I'm assuming it's a block cipher as a result of the blocks of output it produces.
You can try to determine the block size by modifying each byte of the input data, starting from the last one.

A typical block cipher implementation will mix the change into all other bytes in the block, and usually also to all following blocks, but not to blocks that come before it.

So if you observe e.g.:

 Plaintext      Encrypted
 0000 0000      2348 1234
 0000 0001      2348 4292
 0000 0010      2348 9823
 0000 0100      2348 2149
 0000 1000      2348 6785
 0001 0000      8173 1437


you could be quite certain that it is a block cipher with 4-byte blocks.

For comparison, in typical applications of a stream cipher, only a single byte would usually change. If it is a stream cipher, you can obtain the keystream by encoding 000...0000, because it sounds like it doesn't have a random initialization vector.