Depending on the size of the project, and the organisation, there will be differing approval processes. These need to be understood in advance. Typically the Charter will need agreement from an Executive Level Committee to proceed.
It is likely that particular aspects may need to be modified before approval. The documents should be amended (new version number created) to reflect changes. The final version should be signed off by the relevant authority. Whilst a signature on the document is preferable, a note confirming that Version 1.x has been approved will suffice.