I've been through A Wrinkle in Time, A Wind in the Door, A Swiftly Tilting Planet and Many Waters, and I can't find any discussion of the mechanism of tessering. Frequent mention is made of the Tesseract but this seems to be more of a concept than anything physical.
When Mr. Murry escapes from the column he doesn't need any equipment to do it. He does have Mrs. Who's glasses, but it isn't clear whether it's these that allow him to tesser or whether they just allow him to see where to tesser to. I assume it's the latter since he obviously didn't have the glasses when he first tessered out of the laboratory on Earth.
Mrs. Who, Which and Whatsit can tesser without any apparent equipment.
I suspect L'Engle was simply not interested in the hard Sci-Fi aspects of tessering and I don't think we'll get a definitive answer from the books. Unless there are interviews with L'Engle or similar out there in Googlespace I suspect there is no answer.