Probably, but there's no Adventure-canon answer.
Before expanding on that, consider that digimon don't necessarily have a single digivolution path. Even within the Adventure canon, we've seen examples of the the same partner digimon being able to digivolve into different forms - specifically, SkullGreymon instead of MetalGreymon in the original series, MetalGreymon (Virus) instead of MetalGreymon in the 02 series, and Ophanimon instead of Magnadramon in Tri. So it's probably not fair to say that the partner digimon have "personal" ultimate forms as much as it might be "preferred" or "typical."
The digivices "serve to channel the emotional energy of the DigiDestined, allowing their Digimon partners to digivolve." (source: Digimon Wiki) When this emotional energy is expended, the digimon reverts back to their standard form (or lower, if a particularly large amount of energy was used), and even this standard form is not consistently the same level, such as Gatomon being Champion-level in day-to-day activities.
So in some sense, digimon do digivolve on their own based on some form of age, experience, or the like, which explains the majority of non-partner higher form digimon. The digivice then allows partner digimon to digivolve further temporarily. As life forms made of data, this is typically explained based on whether they've "downloaded" enough data, which makes them stronger.
While it's not clear what specifically determines which form is used, the examples above all involve unusual circumstances as compared to when each normally digivolves. SkullGreymon comes about when Tai attempts to force Greymon's digivolution in an improper show of Courage. MetalGreymon (Virus) is forced by the controlling energy of a Dark Spiral. Ophanimon results from Kari's emotional breakdown during the closing scenes of Movie 5 in Tri.
When we consider other sources of canon, such as the video games, we see much more branching digivolution trees. If we consider Digimon: Cyber Sleuth, as a recent example, Veemon has six different forms it can digivolve in to, depending on stats and levels: the Champion forms ExVeemon, Veedramon, Lobomon, and Strikedramon; and the Armor forms Flamedramon and Magnamon when in possession of the relevant DigiEggs. Notably, in the case of Magnamon, even this is not exclusively achievable by Veemon. Guardromon (Gold) also can digivolve to this form in the game.
So while there isn't a canon answer for what Davis's ExVeemon would further digivolve into independently, if it could via a digivice, certainly an ExVeemon does have the ability to digivolve to Ultimate form, given the appropriate time and energy, and the same for an Aquilamon and an Ankylomon. There is no Adventure-canon answer for what those forms would be, however.
