There's some terminology confusion here. In conventional parlance you don't "stain" wood light. Stain only alters the basic colouration of wood1 or makes it darker, or both. Although there are various ways to colour wood a little lighter (including just using diluted white paint!) this isn't technically staining.
So, my question is: Can red oak be bleached first and then stained to look more like white oak?
Yes you can bleach all woods to make them lighter, using any of the two-part or A+B bleaches2. With a fairly typical red oak I'm not sure if you would need to stain it after to make it look like white oak, it could already be within the correct colour range. Obviously this is something that would need some experimentation, and it might vary with your wood which leads on to the next point.....
What colour is each anyway?
All wood varies to begin with (sometimes quite a lot) so we're never just dealing with a single colour no matter what the wood is. And there's heartwood v sapwood, the sapwood usually being lighter, sometimes much lighter as in walnut. Additionally, the dividing line between what's sold commercially as white or red oak isn't nearly as clear as the books and endless magazine articles might lead us to believe!
In reality there is no one White Oak or Red Oak, but various subspecies that are assigned to two categories (see the Wood Database and other resources for more). And some are much less characteristic of their supposed colouring/features than one would expect3.
So in short, it's possible to find some red oak that's already quite similar to white oak — within the range of colouring that white oak naturally exhibits — so you wouldn't have to do anything to it at all.
I presume you're in the US or Canada so this probably isn't an option since you lost virtually all your chestnuts to a blight last century, but elsewhere in the world chestnut would be a viable and possibly much cheaper alternative. It's long been referred to as "poor man's oak" it can look so similar, although it lacks oak's striking quarter-sawn figure or ray flecking (which obviously can be beneficial in some applications)
Thanks to @gnicko from the Comments:
While it's often stated that this is a close match or good substitute for American ash, as well as chestnut (the Wood Database entry mentions both), you can clearly see in images online that it can also look a heck of a lot like typical white oak. I would expect some regional variation on the availability of sassafras so you'd need to check local mills and/or hardwood dealers to see if they stock it.
1 Changes the hue — more reddish, yellowish etc.
2 Sodium hydroxide and hydrogen peroxide, mixed together or used in sequence. Use with caution!
3 Despite how confident most of us are that we can tell which is which on sight, this is why some hardwood dealers and wood experts won't ID a piece of oak as one or the other unless they have it in front of them and can examine the prepared end grain with a loupe or microscope O_O