Does any online site refer to evidence that this monk ever existed as an actual historical figure? If the person is believed to be so important and famous as next Buddha then where is the tomb?
Many Chinese Buddhists believe he was a reincarnation of Maitreya, the person who will become the next Buddha after Gautama Buddha.
According to the non-simple version of Wikipedia,
The main textual evidence pointing to Budai resides in a collection of Zen Buddhist monks' biographies known as The Transmission of the Lamp.
i.e. that is the historical evidence.
See also The Jingde Record of the Transmission of the Lamp -- I guess that's not translated but I think you can read Chinese so you might like to read for yourself what it says about Budai, there.
Incidentally, "history" is defined as text -- i.e. everything we know about history comes from historical texts, by definition. The time before our earliest recorded/surviving texts is called "pre-history" or "pre-historic". The time of the Buddha (500 BCE in India) is sort of pre-historic, except that we have the Tipitaka and so on transmitted via "oral tradition".
Apart from texts received from the past, there's the study of old things. That's a separate topic, i.e. "archaeology". Pre-historic times are known only through archaeology. People study historic times using history and archaeology.
As an example of the archaeology of early history, we have historic texts about the Roman war with Carthage, plus people go to dig in Rome and Carthage to see what they can find, the remains of city walls and houses and temples etc. Usually, in my experience, stone and pottery are the only materials which survive (intact or broken) to be dug up, every other material tends to decay -- and even stone doesn't survive, was already previously dug up and scattered or reused, on land on which people have built new things, unless the older archaeological layers are buried underneath -- and graveyards, or any place like that, tend to fill up and be recycled within a few hundred years.
Or sometimes something is preserved as a valued relic by or in a temple (although 1000-year-old temples aren't very common either, the Potala is only 500 years old).
The word Jingde (景德), the first two characters of the title, refers to the Song dynasty reign name, which dates the work to between 1004 and 1007 CE.
Our detailed knowledge of history is mostly from historical sources -- not from archaeological sources nor contemporary monuments. Archaeological evidence is rare i.e. you're lucky to find anything that old (even if something is made of stone, for example, later people/civilisations often reuse those old stones to build something new). So if you're looking for archaeological evidence, an individual tomb from 1000 years ago, you may be disappointed. Perhaps you also won't find tombs of specific kings from that period. Also I don't know what the burial customs of that time and culture were, perhaps it was cremation without necessarily preserving/venerating the remains (plus I haven't read what The Transmission of the Lamp says, if anything, about the circumstances of Budai's end-of-life).
I'm not even sure what archaeological evidence there is for the Buddha's existence (I say that to point out the limits of archaeology, not to doubt the Buddha's existence). There are some present-day objects which, tradition says, are relics (e.g. a tooth or several teeth) of the Buddha. One of the earliest remaining artefacts are the Pillars of Ashoka -- I guess we're lucky to have them, and you can see from that the kind of thing which might survive for millennia -- i.e. imperial monuments more likely than individual tombs. Some tombs survive too, but by chance.
Importantly the absence of a present-day tomb isn't proof of the non-existence of a historical person.
There is research and writing to proof Budai monk is a historical figure around 10th century and under category of Buddhism Zen teaching.
The source found that seems not having enough of a specified quality and could not be suitable to Junior Buddhist.
"The sources available to modern scholars concerning Pu-tai' s life are deficient". The materials give merely a nebulousaccount, all are agreed that he was a mysterious cleric whoperformed miracles.
Reference
https://wikivisually.com/wiki/Budai
Hsiao, Bea-Hui (1995) Two images of Maitreya: Fu Hsi and Pu-tai Ho-shang. PhD thesis. SOAS University of London. https://eprints.soas.ac.uk/28925/
