To your question. Yes. What you propose is typically called the back door Roth. You make the (non-deductible) IRA deposit, and soon after, convert to Roth. As long as you have no other existing IRA, the process is simple, and actually a loophole that's still open.
If you have an existing IRA, the conversion may be partially taxed based on untaxed balance. As comments frequently get overlooked, I'm adding @DilipSarwate excellent warning regarding this -
Depending on the value of the existing Traditional IRA and its
pre-existing basis, if any, the backdoor Roth conversion might be
almost completely taxable. Example: Traditional IRA worth $250K with
zero basis. New nondeductible contribution increase value to $255.5K
and basis $5.5K. Converting $5.5K into a Roth IRA leaves $250K in the
Traditional IRA with basis $5381.60. That is, of that $5500
conversion, only $118.40 was nontaxable and so, not only is the
original $5500 taxable income to the OP but he also owes taxes on
$5381.60 of that $5.5K conversion.
In short, discussions of backdoor Roth conversions as a great idea should always be tempered with an acknowledgement that it does not work very well if there is any other money in the Traditional IRA. Once that nondeductible contribution enters a Traditional IRA, it does not come out completely until all your Traditional IRA accounts are drained of all money. All your Traditional IRA money is considered by the IRS to be in a single pot, and you can't set up a Traditional
IRA (possibly with a new custodian) via nondeductible contribution, convert just that Traditional IRA account into a
Roth IRA account, and claim that the whole conversion amount is nontaxable because all the tax-deferred money is in the other IRAs that you haven't touched at all.
Last - you disclosed that you are depositing to a Roth 401(k) to the match. Which prompts me to ask if this is best. If your marginal rate is 25% or higher, you are missing the opportunity to save 'off the top', at that rate, and 'fill the lower brackets' at retirement, or, via conversion, any year before then when you are in a lower bracket for whatever reason. See my answer for Saving for retirement: How much is enough? which addresses this further.
From new comments -
Won't his Roth 401k contributions max out his overall Roth
No. They are separate numbers, each with own annual limits.
Wouldn't this prevent any back-door Roth conversions?
The 401(k) has no effect on back door Roth, except for the fact that the 401(k) and high income make the Roth IRA unavailable by normal deposit. Back door is the only door.
At the end are you encouraging him to look for a Traditional 401(k) at
work to max out, then contributing to a Roth?
Yes! Read the linked SE article, and consider the annual withdrawal that would get you to 25%. As I wrote, it would take $2M+ to 'fill' the 15% bracket at retirement.