Christopher Lythgoe, 31, is charged with granting a fellow member of the banned group permission to carry out the attack on July 1 this year.
A 22-year-old man from Lancashire, who cannot be named for legal reasons, is charged with intending to commit acts of terrorism contrary to Section 5 of the Terrorism Act 2006.
The charge states that he bought a "Gladius Machete" for the purpose of murdering West Lancashire MP Ms Cooper between June 5 and July 3 this year.
He is also accused of making threats to kill a female police officer on July 1.
Lythgoe, of Warrington, Cheshire, is charged with encouraging the 22-year-old to commit murder and both men are also charged with being members of National Action.
They appeared in the dock at Westminster Magistrates' Court in London on Friday alongside four other alleged members of the banned group.
They are Garron Helm, 24, of Seaforth, Merseyside; Matthew Hankinson, 23, of Newton-le-Willows, Merseyside; Andrew Clarke, 33, of Warrington; and Michael Trubini, 35, also of Warrington.
The six men are accused of continuing to be members of National Action after the group was banned under UK law on December 16 last year until their arrest on September 27 this year.
They spoke to confirm their names, dates of birth and ages.
During the hearing, Helm asked: "Am I allowed to speak?" before he was approached by his lawyer.
Chief Magistrate Emma Arbuthnot remanded them in custody before a pre-trial hearing at the Old Bailey on Friday, November 3.
The suspects were among 11 men arrested by counter-terrorism police in raids across the UK last month.
In a statement on Thursday, released after the men were charged, Ms Cooper said: "I would like to thank everyone involved in this case, especially the counter-terrorism police, for keeping me, my staff and the public safe.
"There remains an ongoing criminal investigation so it would not be appropriate for me to comment further."
National Action became the first extreme right-wing group to be banned under terrorism laws in December 2016.
The proscription meant that being a member of or inviting support for the organisation is a criminal offence carrying a sentence of up to 10 years' imprisonment.
An entry for National Action in the official list of proscribed groups says it is a "racist neo-Nazi group" which was established in 2013 and has branches across the UK that "conduct provocative street demonstrations and stunts aimed at intimidating local communities".
The document also links National Action to the murder of Labour MP Jo Cox in 2016.
It said the group's online propaganda material frequently features extremely violent imagery and language, and cited tweets posted in connection with her murder at the hands of right-wing extremist Thomas Mair.