Is the IRS likely to reject an installment plan based on their failure to file 2006 in a timely fashion?

If there's anyone left at IRS with the first lick of sense, yes.

Should they send in the extension, and quickly finish both years while waiting for the bill from the IRS?

Don't tell them (they can't be trusted with this information), but there won't be a bill until they file a return. Just tell them to file the extension and get their butts in gear. When they get a notice on either year they can then get in touch with IRS about arranging payment for both.

Meanwhile, 2008 is upon us. Their first estimated tax payment is due tomorrow. Make sure they've made it before they talk to IRS, and make sure they make all their 2008 ES payments. Otherwise, aside from perpetuating this mess, their payment agreement will be in default when they don't timely pay their 2008 liability.

Phil