Do Catholic Churches have Antidoron?
http://www.orthodoxwiki.org/Antidoron
Antidoron (from Greek, meaning "instead of the gifts") is the remaining bread from a loaf of prosphora after the Lamb has been removed for the Holy Eucharist. It is blessed immediately after the epiclesis during the Divine Liturgy and is given by the priest to the faithful after the service.
Prosphora (Greek for "offering") is bread prepared for use in the Divine Liturgy. A portion of it, known as the lamb (or amnon) is cut out during the proskomedia which is consecrated during the Divine Liturgy to be the Eucharist, while the rest is cut up for the antidoron, the blessed bread distributed at the end of the liturgy.
Catholics use unleavend bread. There is no analog to cutting the center "Lamb" out of the bread. Our Eucharistic bread is in the form of flat disks and all is consecrated during the Eucharistic prayer.
SD
From the definition, it looks like what you are talking about is consecrated hosts. Yes, these are kept in the tabernacle.