Margaret Walkling, 82, had placed some artificial mauve carnations in a vase to honour her deceased niece and nephew, but was upset to find that they had disappeared.
The Sandy resident visited the cemetery last Thursday (July 21), searching the grounds when she discovered the items were missing, but to no avail.
Margaret told the Chronicle: "I'm just so angry that someone could do that to a grave, let alone one for newborn babies. What did they get for it?
"They are people with no respect for anything and they certainly don't deserve to prosper."
Margaret's nephew and niece died in 1952 and 1954 respectively, and since her sister passed away, Margaret has been looking after the babies' grave.
She now wishes to warn the public and asks them to check their relatives' plots in case anything is missing, and advises people to stay vigilant in the future.
Margaret added: "What if it's happened to someone else's family and they don't know anything about it?
"I haven't reported it to the police. We don't know exactly when it happened and I walked around [the cemetery] for a time and couldn't find anything.
"They'd just picked up the vase and taken it away. For someone to stoop so low..."
A Sandy Town Council spokesman said: "This is the first time we have been made aware of this situation and we are saddened to hear that a theft of this nature may have taken place".