So, reading between the lines a little, you have a twitter account and a password because this is internal, so you don't want to go with an auth process that requires a user to interact with it?
The idea behind OAuth is that you don't ever find out what the user's password is; I agree that if I'm right about what you are trying to do that it isn't the right thing. The OOB Flow suggested by JohnD has the same problem.
If you do have an account/password, then you can work with submissions to the website directly, using the login form and the tweet form. Unfortunately this means you don't have access to the API (they nuked basic authentication via the API last year) -- depending on what you're trying to do that may or may not be a problem.
Edit:
Use OAuth and remember the token. It never expires, according to the twitter API docs, and since you presumably have some limited number of accounts that you care about, you can just jump through the OAuth hoops once for each account and you're done until you need another account. You can even do the whole thing programmatically given the username and password, assuming they don't stick a captcha in there at some point. But I suspect your best bet is to just use OAuth and store the tokens.