Do you contract a disease or a virus? Or either?
Solution 1:
You can contract both disease and virus. Lexico is quite definitive: definition 3 of the verb contract is "Catch or develop (a disease or infectious agent)".
Contract(ing) a disease is certainly common. The OED has "To enter into, bring upon oneself (involuntarily), incur, catch, acquire, become infected with (something noxious, as disease, †mischief; bad habits or condition; †danger, †risk, †blame, guilt)." (with daggers indicating obsolete usages.) Examples go back to the 17th century e.g. "Thereby contracting dangerous Colds, Coughs and Catarrhs." (contract, v. OED Online) This is also found in modern scientific and general sources e.g. "Almost half of children who contract covid-19 may have lasting symptoms" (New Sci., 2021), "Dutch Minks Contract COVID-19" (NPR, 2020)
Contract a virus is also widely found. Collins Cobuild has some examples, e.g. "Hundreds of staff in 20 hospitals have been randomly tested as part of efforts to understand if they are contracting the virus from patients." (The Sunday Times, UK) "Imposing stricter restrictions in areas where people are at a low risk of contracting the virus would do unnecessary damage." (The Sun, UK) Lexico cited above gives "three people contracted a killer virus" as an example.