any statistics on how much tipped employees under-report income

I don't know.

For restaurants where tipping is the norm, an employer has met his responsibility to assure tips are reported, and is not required to allocate tips, when at least 8% of the tab has been reported as tips.

Any data from the IRS would probably be buried in the reports they issue on the Tax Gap. You may want to revise your search to Tax Gap and Unreported Tip Income.

Well then, I would guess the underreporting percentage at 7%.

statistics on how much tipped employees under-report

Thanks for the keyword "tip income". Adding that to my search I came up with

An IRS study showed that the amount of tip income reported in CY 1993 was less than one-half of the tip income, leaving over $9 billion unreported.

Does CY stand for calendar year?

Anyway, I'd like a study that is more up to date. Since 1993 there have been new tip agreements to increase compliance, so maybe the tax gap on unreported tip income is smaller now.

Also, does the $9 billion estimate only cover the restaurant industry? What about coffee shops, movers, strippers, travel guides, etc?