There is nothing wrong with giving the entire fidyah to one poor person

The one who is unable to fast on an ongoing basis is obliged to feed one poor person for each day that he does not fast, because Allah says (interpretation of the meaning): 

“And as for those who can fast with difficulty, (e.g. an old man), they have (a choice either to fast or) to feed a Miskeen (poor person) (for every day)” [al-Baqarah 2:18 ]

Ibn ‘Abbaas (may Allah be pleased with him) said: This has not been abrogated. It refers to an old man or old woman who is not able to fast; they should feed one poor person for each day. Narrated by al-Bukhaari, 4505. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

“With regard to how this feeding is to be done, it may be done in two ways: 

1 – Making food and inviting the poor to come and eat, according to the number of days that one has to cover, as Anas ibn Maalik (may Allah be pleased with him) used to do when he grew old. 

2 – Giving them uncooked food. 

Al-Sharh al-Mumti’, 6/335. 

See also question no. 49944. 

With regard to feeding one poor person for thirty days, many scholars have stated that this is permissible. This is the view of the Shaafa’i and Hanbali madhhabs and of a group of Maalikis. It says in al-Insaaf (3/291): It is permissible to give food to one poor person all in one go. 

See Tuhfat al-Muhtaaj, 3/446; Kashshaaf al-Qinaa’, 2/313. 

It says in Fatawa al-Lajnah al-Daa’imah (10/198): 

Since the doctors have ruled that this sickness that you are suffering from is ongoing, and you cannot fast as a result, and there is no hope of recovery, then you have to feed one poor person for each day, giving half a saa’ of the local staple food, whether it is dates or something else. If you give the poor person dinner or breakfast for the number of days that you owe, that will be sufficient. 

From this it is known that it is permissible to feed one poor person for thirty days, or to gather thirty poor persons on one day and feed them. And Allah knows best.