Morfin likely did not attend Hogwarts.
Considering the personality of Marvolo, Morfin’s father, it seems incredibly unlikely that he would have sent his children to Hogwarts. Sending Morfin and Merope to Hogwarts would mean they would be among several wizards with lower blood statuses than their own. From his reaction to Bob Ogden’s arrival, it seems clear he avoids interaction with lower blood statuses.
“Should’ve made your presence known, shouldn’t you?’ said Gaunt aggressively. ‘This is private property. Can’t just walk in here and not expect my son to defend himself.’
‘Defend himself against what, man?’ said Ogden, clambering back to his feet.
‘Busybodies. Intruders. Muggles and filth.”
- Harry Potter and the Half-Blood Prince, Chapter 10 (The House of Gaunt)
Marvolo also held the Ministry of Magic in very low esteem, since he did not consider them as having the authority to summon his pure-blooded son to a hearing for his crimes. This further implies that he would not want to send his children to attend Hogwarts, where they as students of the school would be subject to the authority of teachers and staff who were not pure-blooded.
“Summons! Summons? Who do you think you are, summoning my son anywhere?’
‘I’m Head of the Magical Law Enforcement Squad,’ said Ogden.
‘And you think we’re scum, do you?’ screamed Gaunt, advancing on Ogden now, with a dirty yellow-nailed finger pointing at his chest. ‘Scum who’ll come running when the Ministry tells ’em to? Do you know who you’re talking to, you filthy little Mudblood, do you?”
- Harry Potter and the Half-Blood Prince, Chapter 10 (The House of Gaunt)
Marvolo did not even open letters he received by owl - which almost certainly would all sent by wizards, as very few Muggles would be capable of sending letters by owl.
“Yes, Mr Gaunt. I’ve already told you. I’m here about Morfin. We sent an owl –’
‘I’ve no use for owls,’ said Gaunt. ‘I don’t open letters.’
‘Then you can hardly complain that you get no warning of visitors,’ said Ogden tartly.”
- Harry Potter and the Half-Blood Prince, Chapter 10 (The House of Gaunt)
Since Marvolo did not wish for himself or his children to associate with anyone of a lower blood status than their own or be under their authority, it seems highly unlikely that he would choose to send his children to Hogwarts, where both were sureties. Additionally, it seems likely that Marvolo never attended Hogwarts himself, as Voldemort only found him after researching in old books of wizarding families. Presumably if there were records of him at Hogwarts, Voldemort would have able to find him through those instead.
“All he had to go upon was the single name “Marvolo”, which he knew from those who ran the orphanage had been his mother’s father’s name. Finally, after painstaking research through old books of wizarding families, he discovered the existence of Slytherin’s surviving line. In the summer of his sixteenth year, he left the orphanage to which he returned annually and set off to find his Gaunt relatives.”
- Harry Potter and the Half-Blood Prince, Chapter 17 (A Sluggish Memory)
There is precedent for a Gaunt choosing not to send the child in their care to Hogwarts - Gormlaith Gaunt chose not to send her niece Isolt to Hogwarts, which was a place far too egalitarian and full of Mudbloods for her taste.
Gormlaith refused to allow Isolt to take up her place at Hogwarts when the letter arrived, on the basis that Isolt would learn more at home than at a dangerously egalitarian establishment full of Mudbloods. However, Gormlaith herself had attended Hogwarts, and told Isolt a great deal about the school. In the main, she did this to denigrate the place, lamenting that Salazar Slytherin’s plans for the purity of wizardkind had not been fulfilled.
- Ilvermorny School of Witchcraft and Wizardry (Wizarding World website)
Like Gormlaith before him, Marvolo wanted to ensure that his children grew up believing that they should only associate with pure-bloods. Sending them to Hogwarts would ensure the opposite - they would be forced to interact with wizards of every blood status daily while there. Therefore, it seems most likely that Marvolo did not send either of his children to Hogwarts.