Parliamentary Procedure/Order of speakers in a debate
The chair recognizes speakers in debating an issue and alternates between speakers who are for and against. The chair recognizes the first hand up among those for and then the first hand up among those against. Is there an RONR rule where those who desire to speak register with the parliamentarian or secretary and the P/S gives the names in order received to the chair?
There is no such rule in Robert's Rules of Order Newly revised although the organization can adopt a special rule of order to make this a standard procedure for the organization. However - see my 'However' remarks near the bottom.
In large conventions alternating between for and against is usually handled by having two (or more) microphones on stands - one for those in favor and one for those against. If you don't have (or need) microphones you can designate two different speaking spots/lecterns.
In public hearings (as in legislative bodies - town councils, country board of supervisors) there is often a sign up sheet and this is handled by the secretary. The chair will call off the names of those on the sheet.
As a parliamentarian I often aid the chair in letting the chair know who wants to speak but I would not want to be the keeper of the roll of who wants to talk as there are more important things for the parliamentarian to do.
Robert's Rules discourages a 'lining up' of speakers as it seems your organization may want to do -- such 'lining up' tends to lead to inefficient debate. Having speakers 'sign up' to talk before debate gets going - seldom leads to good debate - instead speakers should respond off of previous ideas/comments. Once signed up in advance many speakers will feel they have to talk even though their point was made by an earlier speaker. This redundancy is inefficient. Although it is of some value to allow everybody to have their own say - it wastes the meeting time and wastes the time of everybody else in the meeting.
Under Robert's Rules, once a speaker finishes, the process of the chair selecting the next speaker starts all over again - everybody has a chance to be the next speaker (if they haven't spoken previously.) The selection rules, such as alternating between pro and con, are guidelines for the chair who has some discretion in picking the speakers - with the aim of keeping the debate moving towards disposal of the motion.
A meeting is primarily about debate on an issue and then having the organization taking an action or a position. Debate in a meeting is not about 'free speech' nor is it about the individual.
Therefore unless there is a good reason (such as taking of public comment) I would advise against a procedure that 'lines up' speakers in advance. It seldom leads to debating the issue in a logical way. If you have a good reason to make up a standard rule such as you suggest, the organization can do it, but the consequences tend to result in a less efficient process.