This term originated not in any figurative sense, but literally - both bibs and tuckers were items of women's clothing from the 17th to late 19th centuries.
Early bibs were somewhat like modern day bibs, although they weren't specifically used to protect clothes from spilled food as they are now. Tuckers were lace pieces fitted over the bodice - sometimes called 'pinners' or 'modesty pieces'. These were known by the late 17th century and were described by Randle Holme in The Academy of Armory, or a Storehouse of Armory and Blazon, 1688:
"A Pinner or Tucker, is a narrow piece of Cloth - which compasseth the top of a Womans Gown about the Neck part."
Tuckers, as the name suggests, were originally tucked in. Pinners differed by being pinned rather than tucked. Pinner is clearly the precursor of pinafore - originally pin-a-fore, i.e. pinned on the front.
Incidentally, the blazons of the title of Holme's book gave the name to another form of dress - the blazer. Blazons were the heraldic coats of arms or badges of office worn by the king's messenger. Blazer jackets, which became fashionable in the early 20th century as uniforms for supporters of sports teams and as school uniforms, mimicked the heraldic style.
'Best bib and tucker' is an 18th century term, the first known citation of which is from a translation of the Marquis d'Argens' ambitiously entitled work New Memoirs establishing a True Knowledge of Mankind, 1747:
"The Country-woman minds nothing on Sundays so much as her best Bib and Tucker."
Tuckers continued to be worn until the late 19th century. Charlotte Bronte referred to the practice in Jane Eyre, 1847:
"Some of the girls have two clean tuckers in the week; the rules limit them to one."
'Tuck' is a slang term for food which was coined in English public schools in the 19th century. For example, Thomas Hughes' Tom Brown's School Days, 1857:
"The Slogger looks rather sodden, as if he didn't take much exercise and ate too much tuck."
This migrated to Australia, where it was modified to tucker. Both this meaning of tucker and the women's bib meaning have connections with food and it is tempting to speculate that they are in some way connected. It seems that they aren't. Tucker in the food sense derives from the earlier term 'a tuck-out' (later also 'tuck-in'), which meant 'a hearty meal'. 'Tuck-out' was synonymous with 'blow-out'. Both terms are listed in John Badcock's Slang: A Dictionary of the Turf, 1823:
Blow-out - a good dinner will blow-out a man's tripes like any thing; so will a heavy supper. Either, or any other gormandising meal, is also 'a famous tuck-out'.
'Blow-out', which appears to have had quite a crude meaning, is a long way removed from the protective crinoline bibs worn by Jane Austen heroines.
Another link that is sometimes made is the possible connection between tucker and tuxedo. The two names sound similar of course and the cummerbund that is usually worn with the formal tuxedo suit is rather like a tucker. There's no foundation to that notion. Tuxedos are named from Tuxedo Park, New York, where they were first worn in 1886.