which early sources mention them?
Pliny the Elder, Tacitus and Ptolemy.
Vandals
The earliest mention of the Vandals is from Pliny the Elder, who used the term "Vandili" in a broad way to define one of the major groupings of all Germanic peoples. Tribes within this category who he mentions are the Burgundiones, Varini, Carini (otherwise unknown), and the Gutones.

Tacitus mentioned the "Vandilii", but only in a passage explaining legends about the origins of the Germanic peoples. He names them as one of the groups sometimes thought to be one of the oldest divisions of these peoples, along with the Marsi, Gambrivii, Suebi but does not say where they live, or which peoples are within this category. On the other hand, Tacitus and Ptolemy give information about the position of Varini, Burgundians, and Gutones in this period, and these indications suggest that the Vandals in this period lived between the Oder and Vistula rivers.

Ptolemy furthermore mentioned the Silingi who were later counted as Vandals, as living south of the Semnones, who were Suebians living on the 
which emperor loved greek culture and adopted the wearing of a beard UNANSWERABLE where did caligula reign/ Rome. during which time? AD 37 to 41. who was his predecessor? Tiberius. who plotted against him? Cassius Chaerea. whom did he marry? UNANSWERABLE did he take part in any battle? Yes, during his reign, the empire annexed the client kingdom of Mauretania as a province. where is this located? It stretched from central present-day Algeria westwards to the Atlantic, covering northern Morocco, and southward to the Atlas Mountains. when did the vandals capture this place? 420s where were these invaders from? Southern Poland. how are they classified? They were not considered "Germani" by ancient Roman authors but they are classified as a Germanic people by modern scholars.
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Vandals [SEP] Early classical sources
The earliest mention of the Vandals is from Pliny the Elder, who used the term "Vandili" in a broad way to define one of the major groupings of all Germanic peoples. Tribes within this category who he mentions are the Burgundiones, Varini, Carini (otherwise unknown), and the Gutones. Tacitus mentioned the "Vandilii", but only in a passage explaining legends about the origins of the Germanic peoples. He names them as one of the groups sometimes thought to be one of the oldest divisions of these peoples, along with the Marsi, Gambrivii, Suebi but does not say where they live, or which peoples are within this category.