Is it permissible to go for the obligatory Hajj or Umrah by installments, and does this make the person capable, and if it is not permissible, then who is sinful?

One is not obligated to seek out expenses for 'Umrah or Hajj, nor to borrow money to perform them. Rather, the pilgrimage is obligatory only for those who are financially and physically able. It is permissible to accept a benevolent loan (qardh hasan) or a legitimate financing arrangement to perform the pilgrimage for someone who is certain of their ability to repay, as stated by Al-Khatib Al-Shirbini.