Both defendants deny the charges and have been remanded on bail.
Charging documents allege the pair deposited $746,881 between September 30, 2015, and February 14 last year with "intent to deceive the donors of the monies, the party secretary of the New Zealand First Party and/or the Electoral Commission".
In December, Judge Deidre Orchard in the North Shore District Court dismissed an application for continued suppression by the first defendant.
However, an appeal was filed with the High Court, which keeps the suppression orders in place until a ruling on the challenge.
Suppression has been opposed by the SFO and a consortium of media organisations, including the Herald's publisher NZME, which unsuccessfully attempted to name the pair before last year's election.
The NZ First Party also attempted to stop the charges becoming public until after a government was formed.
The first defendant has complained the case has been "politicised" and is an attack on NZ First leader and former Deputy Prime Minister Winston Peters.
Judge Orchard, however, said in her decision the case is inherently political.
Peters has distanced himself from the foundation - reported to have bankrolled the political party - and has denied any wrongdoing after it first came under scrutiny in November 2019.
After the charges became public, Peters claimed at a press conference he and the party had been "exonerated".