You won't be putting them on the spot by calling them up and asking to speak with someone, providing you're polite, well-mannered and patient. Ultimately you need to know the right person to write to so why not introduce yourself at the same time and make whoever you speak with a promise to send your CV in, if that's the right thing to do.
You can then follow up the conversation with your CV and covering letter, proving that you're true to your word and are keen to take things further. Hopefully you'll be seen as someone who not only has the initiative to approach them looking for a job but is also keen to impress.
This all assumes you can get past the receptionist, again assuming they have one, and the key to doing that is to be exactly unlike everyone else that calls up to try and get through to someone they've never spoken to before. Be polite, patient and professional and above all - ask nicely! If you get brushed off take it with good grace, but don't put the phone down without a contact name and address details for your CV. You might even get an email address, which you could use, but I'd still send a letter in.
Hell, you could even hand deliver it.