Do it yourself? Sure you can!
While outsourcing can save a lot of time tracking credit eligibility if you hire a couple of dozen or more new employees, if you only hire a handful of new people each year, it can be worthwhile doing the work yourself. The time your HR people spend on the phone getting help from the IRS, and from the Department of Labor will usually be more than offset by the tax credits you end up claiming.
If you have the time to make sure that you have all the up-to-date paperwork. You'll need to be educated about what makes your company eligible for specific credits, whether multiple credits can be claimed under specific circumstances, what conditions would disqualify you from claiming certain credits—and there's a whole lot more you'd need to know.
The fact is that unless you have a background in this area, your time is probably better spent working on other areas of your business and having an expert deal with this. After all, it's taxes—and there are few surer ways to undermine your business than by filing inaccurate taxes. For example, if you apply credits where none are eligible, and underpay your company's taxes by $60,000 for the year, the IRS will demand the $60,000 underpayment and apply an additional penalty of $6,000 for the significant underpayment. And that's the best case. If they believe that your company's underpayment constitutes fraud, that fine will be bumped up to as much as $45,000. There are all kinds of penalties that the IRS can impose, and you'll need to be aware of them, and what to do to avoid making a mistake on your taxes and having to pay them.
So yes, you can do this yourself. But if you have more than about 30 eligible new hires, you probably want to look at outsourcing.
If you decide to handle this internally, remember that being too aggressive or too cautious in applying credits can impact your bottom line. It's a thin line to walk, so you should find a contact at your local IRS and Department of Labor offices for when you need to ask questions.