I think a little brush up on Medieval history and the American Van Palt surname might be useful here. The family name was not Van Pelt. The surname was a mistranscription of the Dutch phrase "van Pelt"
About two-hundred years before the Country of Beligum was formed or established as an independent Monarchy it was land held by the HRE and ruled by tbe Princce-Bishop of Liege. In 1622 Anthonius Lanen was gedopt te Sint-Marinus Kerk te OP. zoon van Jois Laen. His six children were baptized at Guilderland as Thunissen or a variant.
In 1663 this man arrined in New Amsterdam with his family. There is nt indication the name Van Pelt was used in any church or legal documents prior to his death circa 1698. However in 1700 there is record showing five Van Pelt children deeded land, belonging to their father Anthonius Lanen, to their brother Aurt Van Pelt. The Lanen family in America has been know by the name Van Pelt since 1700.
In 1701 in Sint-Martinus records there is a marriage of a Vam Pelt . This to me indicates there was a family with the Van Pelt surname in the Pelt area that would ulitimately becom NEBelgium after Napoleon was sent home in defeat in 1815. .
AFAIK the early records areunder the control of the Chief Archivist in Hasselt. Where Van Pelt orfinally came from is speculation. There are certainly still many families of this name in the Flanders Region.
It is quite possible the A. Van Pelt in Chicago is a descendant of Anthonius Laenen as Anthony and the variant spellings was and is still a popular name in this family. The Chicago Library (Main brancb and not Newberry Library) has a long history of Chicago and also the phone books. This where I would begin my search.