You are very close to figuring it out. No, a covalent bond doesn't form between ion pair reagents and the analyte, it is simply a strong association between two charged molecules. Under typical reverse phase conditions, they associate more strongly, forming something like a neutral salt.
Consider the old simple laboratory experiment, where you separate an organic amine, an organic acid and an non-polar organic compound, by shaking an solution of them all in an organic solvent, first with aqueous base (to convert and extract the organic acid into a salt), then with aqueous acid (to convert and extract the organic base.) Ion pair chromatography is something like that, but making a looser association, and making a charged species more non-polar, and instead of shaking, a linear series of extractions down the column.