Regarding intravenous contrast, the type of contrast is different between CT which uses iodinated contrast versus MRI which uses gadolinium contrast. According to the American College of Radiology (ACR) Manual on Contrast Media, there is no cross-reactivity between different classes of contrast medium. For example, a prior reaction to iodinated contrast medium does not predict a future reaction to gadolinium contrast medium, or vice versa.
Regarding oral contrast, the most common agent used is barium sulfate, which is inert; hence, there is no appreciable risk of an allergic reaction (something like 1 in 750,000). However, some institutions use iodinated contrast made for intravenous injection but dilute it and administer it orally; since approximately 1-2% of this is absorbed through the gut, there is a theoretic risk of an allergic reaction. While oral contrast is administered for most CT abdomen/pelvis, it is typically not administered for most MRI abdomen, with some exceptions such as MRI enterography which evaluates the bowel.
Regarding your mother's case, it is statistically most probable her prior allergic reaction (hives) was to intravenous contrast rather than oral contrast. Adrenal pathology can be evaluated with different modalities, including CT, MRI, and PET/CT. For CT, the typical protocol is adrenal protocol CT abdomen with and without contrast, which consists of precontrast, postcontrast at 1 min, and postcontrast at 15 min. For MRI, you can get by with just doing MRI abdomen without contrast, because the key sequence is precontrast in- and out-of-phase which will tell if you if the adrenal nodule is an adrenal adenoma versus not an adrenal adenoma. The addition of contrast on MRI is not essential and will add little additional information. While some pathology such as pheochromocytoma may demonstrate intense enhancement, there are other clues such as "light bulb" bright appearance on MRI T2-weighted sequence as well as clinical and/or laboratory presentation that would raise the possibility of this diagnosis. To sum up, it is probably adequate for your mother to get MRI abdomen without contrast. Alternatively, if there is clinical concern for ruling out metastasis, consider PET/CT; this injects nuclear material for the PET portion but does not inject iodinated contrast for the CT portion (at most institutions).
Thank you providing the reason for evaluating your mother's adrenal glands. The more common indications for performing adrenal protocol are to evaluate an adrenal nodule to determine if it is an adrenal adenoma versus something else such as a metastasis. In these cases, performing the CT with and without contrast and obtaining images at precontrast, postcontrast at 1 min, and postcontrast at 15 min are necessary to calculate the washout. In your mother's case, it sounds like the reason for the study is not to characterize a nodule and/or rule out cancer but rather just to evaluate the evolving morphology of the adrenal glands. Hence, I am not really sure if contrast is that critical, but you can discuss further with the ordering doctor (endocrinologist) and/or interpreting doctor (radiologist) who are more familiar with the details of your mother's case. Your mother's case is very unique, so rare I had never encountered it before and had to search the medical literature on PubMed. Here is one case report that sounds similar to what you are describing:
It basically results in a form of adrenal insufficiency. Adrenal insufficiency presents differently at different stages and is best imaged with CT. In the acute phase, the adrenal glands appear enlarged, heterogeneous, and hyperdense (hemorrhagic). In the subacute phase, the adrenal glands appear enlarged with decreased density (caseation or necrosis). In the chronic phase, the adrenal glands appear chronic, small, and atrophic, with or without calcifications. To sum up, it is probably sufficient to get CT or MRI abdomen without contrast, slightly preferring the former since it can better see calcifications and is less susceptible to patient motion artifact.