Law firms can only be established and owned by attorneys. Non-EEA nationals cannot qualify as an attorney, and hence cannot own law firms either.
"Section 58 
(1) The regional bar association shall, upon request, admit as an attorney-at-law anyone who:
a) is a citizen of any State that is a party to the Treaty on the European Economic Area,
[...]
Section 87 
(1) A law office is a legal entity established by one or more attorneys-at-law or European Community lawyers for the purpose of jointly performing the professional activities of an attorney-at-law as a business-like activity, where a member’s obligation comprises the provision of assets and other services to the law office as determined in the deed of foundation. [...]" (Act on the professional activities of attorneys-at-law (LXXVIII of 2017), subsequently amended, Sections 58 and 87 [unofficial translation as in force on 1 January 2018])
Act on the professional activities of attorneys-at-law (LXXVIII of 2017), last amended 1 July 2021, Sections 58 and 87 
[<a href="https://njt.hu/jogszabaly/2017-78-00-00">Act</a>],[<a href="https://njt.hu/translation/J2017T0078P_20180101_FIN.pdf">Act_Eng</a>]