Rebuke dragons is just for clerics and paladins, but the DM should change that if doing so would be more fun and won't break the campaign
Dragon Magic on Draconic Class Features, in part, says, "Alternative class features replace class features found in the original class description" (11). Then, in the summary of how to read these alternative class features on Class it says that this entry is "[t]he class or classes whose members can select this class feature" (ibid.). The alternative class feature rebuke dragons has the entry Class: Cleric or paladin (14). Thus, technically, there simply aren't any other classes to whom the alternative class feature rebuke dragons is available.
However, bear in mind that with with Third Edition, Wizards of the Coast's assumption with the vast majority of its supplements was that the consumer possessed no other supplements except the core rules and the current supplement and that current supplement had to work without referencing anything but the core rules. Thus, for example, even though the Oriental Adventures shaman at level 3 also gets the supernatural ability turn undead, there's no reason for Dragon Magic to list shaman alongside cleric and paladin as qualifying for the alternative class feature rebuke dragons because Wizards of the Coasts's assumption is that the consumer doesn't have Oriental Adventures.
(This editorial mandate was itself a massive overreaction to Advanced Dungeons & Dragons, Second Edition, a sprawling game without such a mandate that could see a consumer spend good money on a new supplement only for the consumer to get it home, start reading it, and realize he needed several other supplements (some of which might even be out of print) to actually use of the newly purchased product. This led some consumers to accuse TSR (D&D's original publisher) of greed and even malice, something Wizards of the Coast—especially under Adkinson, a lifelong gamer himself—tried to avoid. I'd cite all this, but I don't remember where I saw it; I think it was in an article online or in Dragon on the results of Wizards of the Coast's Third Edition design goals Dragon magazine survey.)
Anyway, with all this in mind, I think it's safe to say that the DM is kind of expected to use his good judgment in such cases… and to look at what else has been published as guidelines. For example, this DM thinks it probably wouldn't hurt most campaigns to allow the Oriental Adventures shaman to take rebuke dragons instead of rebuke undead. Cleric and paladin are both base classes, and the game thinks those classes are okay with it, and it's not like the shaman does anything crazier with turn undead than a cleric does, so another base class having access to the rebuke dragons ability shouldn't deal significant damage.
Nonetheless, Wizards of the Coast, to my knowledge, never published any alternative class features for prestige classes. None. Zero. I don't—can't, really—know why, except that the original purpose of prestige classes were as tools for the DM to better define his campaign, and offering a prestige class like death delver—that're supposed to be intimately familiar with the ways of the deceased, almost deceased, and O-my-God,-it's-deceased-but-it's-walking!—the ability to rebuke dragons is a little weird.
That said, only you know what's balanced for your campaign. There's nothing inherently better about low-level rebuking of dragons that sets off any of this readers' alarms. (For instance, I would worry if there were some way to increase the ability rebuke dragons obscenely and asymmetrically but it seems like there's not or if the player were assembling turning pools for the feat Divine Metamagic and planned to use that feat for ill.) But don't agree to a change that makes you suspicious or uncomfortable!
If making the change doesn't set off any of your alarms either, here's what I'd do: The player wants the class for his PC but the player doesn't want to rebuke undead and wants to rebuke dragons. Fine. Rename the class dragon-death delver and change the ability. Then make dragon-death delvers a thing in the campaign. Maybe they follow around dragons, waiting for them to die? Maybe they sing dirges at dragon funerals? Maybe they defeat the guardians of dragon graveyards and animate looted dragon corpses? Whatever. The player has asked for his PC have something weird; seize the opportunity to do something weird in your campaign.