A most mysterious term, this appears in the middle of the nineteenth century, apparently originally an English dialect term for which no antecedents are known. The English Dialect Dictionary of the end of the century has quocken, to vomit or choke, and quocker, a man who goes harvesting at some distance from home, neither of which is any help at all in explaining a word that means a wooden puppet on a string.
It is recorded best in John Camden Hotten’s A Dictionary of Modern Slang, Cant, and Vulgar Words of 1859:
The term quockerwodger, although referring to a wooden toy figure which jerks its limbs about when pulled by a string, has been supplemented with a political meaning. A pseudo-politician, one whose strings of action are pulled by somebody else, is now often termed a quockerwodger.
Older readers may remember the toy. It was a wooden puppet whose legs and arms were connected only loosely to its body. It was suspended by a single string connected to the head. By jerking the string you could make the puppet flail about in amusing and ridiculous ways. You can see how the political meaning could easily have grown out of that.
Though it is widely recorded in dictionaries of slang in the latter part of the nineteenth century, with Farmer and Henley even describing it as common in Slang and Its Analogues in 1892, and it continues to be included in works on historical slang to the present day, it has rarely appeared in print. The only example I’ve been able to find is in a book of satires edited by William Nation published in 1880: “The shameless arts of the sycophant are not monopolised by Mr. Quocker-wodger and his congeners.”