There is no rule that unambiguously prevents this
The critical rules text from the druid’s Wild Shape feature is (emphasis added):
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.
The dragonborn’s breath weapon is clearly a “feature from your race”, so the question becomes, what is required for a dragonborn to be “physically capable” of using their breath weapon? Let’s look at the description:
Draconic Ancestry. You have draconic ancestry. Choose one type of dragon from the Draconic Ancestry table. Your breath weapon and damage resistance are determined by the dragon type, as shown in the table.
Breath Weapon. You can use your action to exhale destructive energy. Your draconic ancestry determines the size, shape, and damage type of the exhalation.
In the description of the breath weapon, the only explicit requirement is the ability to exhale. Most if not all beasts have this ability. Beyond that, as Jeremy Crawford observes, “no anatomy is specified.” Hence, nothing in the description of the dragonborn’s breath weapon is explicitly incompatible with Wild Shape. If you are content to leave things here, we can answer the question in the affirmative: yes, you can use your breath weapon while wild shaped, because nothing says you can’t.
However, the description of the dragonborn race makes it clear that they are “born of dragons” and that the blood of dragons runs through them. So we can also look for information on draconic breath weapons in the Monster Manual’s section on dragons. The most relevant quote I can find here is:
Dragons are also magical creatures whose innate power fuels their dreaded breath weapons and other preternatural abilities.
(Note: this “innate power” is the kind of background magic that isn’t detected by Detect Magic or negated by Antimagic Field, as described in the Sage Advice Compendium.)
These facts make things a bit more ambiguous. A dragonborn druid that has Wild Shaped into a wolf no longer has the blood of dragons in its veins, since it is now physically a wolf. If you decide that having the blood of dragons in your veins is a physical requirement for a draconic breath weapon, then a wild shaped dragonborn druid would not satisfy this requirement. On the other hand, the fact that draconic breath weapons (and therefore also dragonborn breath weapons) are powered by their innate magical power could be used to further justify the conclusion that the breath weapon is not tied to any specific physical anatomy beyond a mouth with which to exhale the energy.
Therefore, when we look beyond just the text of the dragonborn Breath Weapon feature itself, to the racial description and the description of true dragons, we find indirect support for both positions, but still no clear and unambiguous resolution. So, as with many things in 5th edition, this ultimately comes down to the DM’s ruling. In some sense, this is more of a world-building decision than a rules decision. As a DM, you need to make a decision about how a dragon’s breath works in your world. Is there a specific gland or organ, some part of a dragon’s anatomy and physiology, that powers their breath weapon? Or are they directly manifesting elemental energy through their innate magical power?
In general, I’d recommend allowing things that the rules don’t clearly disallow, especially when there isn’t a huge balance concern in doing so, as is the case here since the dragonborn’s breath weapon is only usable once per rest anyway.