What the law says
LINX does not have a specific article in the Articles of Association that covers this, so the regulation applying is set by the Companies Act 2006 (sections 292,302-5 and 314). This specifies that a group of members controlling 5% or more of the voting rights can either request that a resolution be put to the meeting or require that a resolution be put on the agenda and the same percentage can request the convening of an EGM to address specific business. The same rules apply for notice as normal - 14 days for an Ordinary Resolution and 21 days for a Special Resolution; and 28 days where special Notice is required for say removal of a Director.
The process would be for the representative of this group of members to write formally to either the Chairman or the Company Secretary with the details of the resolution and the members who he or she represents. The clock would start on receipt, but there would clearly need to be a period whilst those details were verified. The 5% of the voting rights would be calculated on the day of receipt even though it might vary by the time of the meeting.
At the time of writing, LINX has about 740 members, but those under common control are combined for voting rights, so in practice there are about 700 members with distinct voting rights. But in essence to table a resolution or to call an EGM and table a resolution, a member would need the support of about another 34 members, each not under common control with any other.
LINX prefers to consult first
Whilst that is what the law demands, and so members ultimately have the right to insist on that timetable, LINX has for some years followed a process to consult at the LINX members' conference preceding the one where the EGM/AGM is scheduled (see https://www.linx.net/about/the-consultation-process-at-linx). Then the motion can be consulted on informally and amended in line with more general feedback to give a good chance of being passed at the formal EGM. The 5% of support would still apply to take it to the consultation, but if the consultation was positive, the Board would then put forward the resolution without further formality.
Let's work together
However, it might be better were the member to discuss this privately with the Board, as the Board can table resolutions rather more simply. Also, with the support of the Board, LINX would consult its corporate (or specialist) lawyers to ensure that the resolution tabled was suitable and would have the effect hoped for if passed. Again, the Chairman or Company Secretary would be happy to ensure such business was brought to the next available Board meeting, or to facilitate a suitable discussion between meetings. This too would first go to an informal consultation at the LINX member conference three months before the EGM or AGM.