When did wildfires start to occur on earth?
A definitive statement comes from the abstract of Scott and Glasspool1, 2006:


  Charcoal, a proxy for fire, occurs in the fossil record from the Late Silurian (≈420 Myr) to the present.


One of the tired old truisms you learn is that fire needs three things: Oxygen, fuel, and a source of ignition.

There is little doubt that there has been lightning since Earth's formation. (Wikipedia:Paleolightning). There appears to be evidence of the fossil fulgurite from the Proterozoic Eon.

Oxygen levels were probably sufficient around 470 Myr. (Wikipedia:Fossil record of fire)

So as soon as plant-based fuel appears on land, there is evidence of fire. 

Although there is evidence for land plants for as far back as 850 Myr, and some evidence for land-based bryophyta (moss) appears in the Ordovician Period, evidence of widespread plant colonization of the land, and plants with vascular tissue that can dry out into fuel, do not appear until the Silurian period with the appearance of Cooksonia and other early tracheophytes. (Wikipedia: Evolutionary history of plants)

The level of oxygen in the atmosphere does appear to play a role, as there is a gap in the Mid-Devonian period, apparently coincident with a dip in oxygen levels (to about 13%), but there doesn't appear to be a consensus on that. At any rate, fossil charcoal picks up again in the late Devonian as oxygen levels rise again. 

1 Andrew C. Scott and Ian J. Glasspool, The diversification of Paleozoic fire systems and fluctuations in atmospheric oxygen concentration,Proc Natl Acad Sci U S A. 2006 Jul 18;103(29):10861-5. Epub 2006 Jul 10.
(Abstract at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1544139/#!po=0.694444)