Yes, but this right is governed by the principle of reciprocity. Under Article 3 of the Law of Real Estate Ownership and Investment by Non-Saudis, foreign diplomatic representatives who are officially approved in the Kingdom may acquire real estate for two specific purposes: their official office premises and the residence of the head of mission and staff members.
The reciprocity principle means that Saudi Arabia extends this right to foreign missions only to the extent that the foreign country in question grants equivalent rights to Saudi diplomatic missions on its own soil. If a country does not allow Saudi embassies to own property, that country's mission in Saudi Arabia may not receive the same benefit under this article.
In practice, this provision is administered through established diplomatic channels and typically does not require individual diplomats to navigate the standard Ministry of Interior application process. If you are a diplomatic staff member with questions about your personal (non-official) property rights, note that those would fall under a different provision — likely Article 2 — and the standard residency and permission requirements would apply.
This is general legal information, not legal advice. For advice on your specific situation, consult a lawyer licensed in Saudi Arabia.