In addition to the Celestial Mount feat or the planar paladin substitution level mentioned in Peregrin Took’s fine answer, there are a couple of other possible options to get a celestial charger companion, though it is not a paladin special mount and does not benefit from that class feature (it could be in addition to a special mount, including a unicorn). On the other hand, these are (maybe) the actual celestial charger, i.e. the one that’s also a cleric 7th.
Leadership—probably, but only in epic
I notice that the celestial charger entry in Monster Manual includes
Level Adjustment [...] +8 (cohort)
The book never explains what “(cohort)” LAs mean, but the general assumption is that these creatures have this level adjustment when chosen as a cohort through the Leadership feat (and that they aren’t otherwise playable). The celestial charger has 8 unicorn HD and 7 cleric levels, so it’s got character level 15th, and an LA of +8 makes it an effective character level of 23.
Leadership says “Regardless of a character’s Leadership score, he can only recruit a cohort who is two or more levels lower than himself,” so in order to recruit an effective character level 23rd creature, the leader must be 25th level, i.e. an epic character. Still, this is an option.
Beloved of Valarian—maybe, and non-epic, but rules are unclear
Another option—maybe—is the beloved of Valarian prestige class from Book of Exalted Deeds. We already have a Q&A about how to become one as a paladin, but this question raises a new issue I hadn’t considered when I answered that question: what exactly does the call celestial charger feature do?
Call Celestial Charger (Su): At 6th level, a beloved of Valarian can call a celestial unicorn—also known as a celestial charger—to her side. See the Unicorn entry in the Monster Manual for the celestial charger’s statistics. [...]
(Book of Exalted Deeds pg. 54)
My initial reading of this feature is that it gets you a celestial unicorn, that is, a unicorn with the celestial creature template applied. That is, of course, what it says here in the outset, you can call a celestial unicorn. Its description of “celestial charger” is that this is just another name for a celestial unicorn. As I said in my other answer, the celestial template doesn’t really do very much for a unicorn—personally I wouldn’t even spend a feat on it, though I would probably be willing to trade remove disease for it.
But Book of Exalted Deeds explicitly says to use the statistics of the celestial charger in the Monster Manual’s Unicorn entry. And that, as Peregrin noted, says
The celestial charger described here is an 8 HD celestial unicorn with seven levels of cleric.
[...] Challenge Rating [...] 13
Beloved of Valarian’s requirements force a character to be 7th level before they can get in (the BAB +7 and Ride 10 ranks requirements each force that minimum), so a 6th-level beloved is, at minimum, a 13th-level character. Which exactly matches the CR of the celestial charger at the level you get it. It may well be that the authors of Book of Exalted Deeds intended for you to get this at this point. Which would be terrible, from an editing stand-point, since the wording on call celestial charger is miserably poor, but Book of Exalted Deeds has a reputation as one of the worst-written, worst-edited books for 3.5e, so this is kind of par for the course.
Now here is the question: is this reasonable? Peregrin opines that the celestial charger would be far too powerful as a special mount, and if we’re talking as just costing an adjustment to your effective paladin level (as with unicorn itself) or a feat (as with Celestial Mount), I’d completely agree with that. But beloved of Valarian costs way, way more than those things—it completely destroys your special mount progression and replaces it with the celestial charger, and it also fails to progress some other paladin stuff. More importantly, it requires three feats just to get in—which for many paladins will be literally all of their feats at this level—and two of those feats are absolute garbage. Plus there’s some really specific roleplay requirements. For all that, its class features are OK, but nothing stellar—I was happy to give them away in a series of feats, and that’s a pretty well-regarded answer.
And honestly, it’s a lot of jumping through hoops. A 7th-level cleric isn’t nearly as strong as a 13th-level cleric, even if it’s also a unicorn and it comes with a quasi-paladin attached.
So this might be reasonable. The celestial charger is probably more powerful than the beloved herself, but frankly that’s kind of the norm for paladins—especially for single-class paladins, the special mount is often the most powerful feature they have. It’s definitely the most optimization-friendly, and you can produce mounts far more powerful than the celestial charger.
The biggest problem is that the celestial charger doesn’t grow at all. At 13th level, it’s pretty solid, but at 20th level it’s looking quite poor.