Congrats on the new baby!

If you go, do be aware that this late in the year, there are often walk-in Whitney day hike and overnight permits available.  Read step 6 in the Wilderness Permit options page (link above on left) for more info. ...or Step 5: Watch the week before, since cancelled permits will show up often.

Regarding a Fire Permit:

Easier than calling and asking, you can download, print and sign one from your computer.  Saltydog provided the links in this thread: Sequoia National Forest Sets New Fire Regulations.  The direct link to the downloadable PDF is here.

Note that it IS called a Campfire Permit, even though it has been written many places that you need one even for a stove.  (I've never been asked for one, and am usually negligent when it comes to carrying one.)  In the White Mountains, there just isn't enough fuel around to be a fire hazard, except maybe at the lower Grandview campground.

By the way, the road to White Mtn is ok, even for a small car, if you don't go too fast.  Just go slow, especially where it is more rocky.  Make SURE you have a properly inflated spare, and maybe carry a can of fix-flat, too.