The founders of Hogwarts had funds from different people but did not have funding from any form of government.
The Wizard card of Helga Hufflepuff as per old JKR website is a as follows,
"[s]he brought people from different walks of life together to help build Hogwarts and was loved for her charming ways."
As per the establishment of Ministry of Magic, it is termed as 1707 in pottermore
The Ministry of Magic was formally established in 1707 with the appointment of the very first man to hold the title 'Minister for Magic', Ulick Gamp.
Prior to 1707, the Wizards' Council was the longest serving (though not the only) body to govern the magical community in Britain. After the imposition of the International Statute of Secrecy in 1692, however, the wizarding community needed a more highly structured, organized and more complex governing structure than they had hitherto used, to support, regulate and communicate with a community in hiding. Only witches and wizards who enjoyed the title of 'Minister for Magic' are included in this entry.
The establishment of The Wizards Council is unclear but it was not mentioned before 1269 anywhere, so we can assume it is not yet formed during the establishment of Hogwarts, which was in/before 933
The Wizards' Council was the predecessor to the Ministry of Magic (FBx). Barberus Bragge was the Chief of the Council in 1269.
The establishment of Hogwarts was due to the growing mistrust on the part of muggles. So, the founders wouldn't ask the muggle government for soft-funding.
Also, the history of Britain during the establishment of Hogwarts is very disturbed because of Viking invasions, we can assume that during this war between Alfred of Wessex and Vikings, the wizards of England region felt threatened and started moving into secrecy.
Late in the 8th cent., and with increasing severity until the middle of the 9th cent., raiding Vikings (known in English history as Danes) harassed coastal England and finally, in 865, launched a full-scale invasion. They were first effectively checked by King Alfred of Wessex and were with great difficulty confined to the Danelaw, where their leaders divided land among the soldiers for settlement. Alfred's successors conquered the Danelaw to form a united England, but new Danish invasions late in the 10th cent. overcame ineffective resistance . The Dane Canute ruled all England by 1016. At the expiration of the Scandinavian line in 1042, the Wessex dynasty (see Edward the Confessor) regained the throne. The conquest of England in 1066 by William, duke of Normandy (William I of England), ended the Anglo-Saxon period.