I would really like to save money during the year and then once a year purchase specific stocks to put in an IRA. Is that possible? Are there companies that offer something like this without charging exorbitant fees?
Absolutely. An IRA works just like any other brokerage account except that there are constraints on contributions and withdrawals. You can invest in just about anything you'd like.
Because the title of your question is slightly ambiguous I'll add this for clarity. You can't directly add stocks to an IRA account. You can only fund it with cash. However, you can buy any stock you want with the cash you transferred into the account. If you want to move a stock into an IRA account from another account you will need to cash it out first, transfer the cash, then re-buy the stock.
You can purchase specific stocks through Fidelity's IRAs for $7.95 per trade.