Question: Is it possible to suffer on the behalf of other sentient beings?
Answer: Most parents, being loving, wish to suffer on behalf of their children when they are sick, but who is able to do this? If this can be done, the world would not have any children who are still sick, or who pass away due to sickness.
There is a Tibetan Buddhist meditation practice called Tonglen, which helps to train in compassion, by visualising taking in all beings’ suffering, and giving them well-being. However, this is for training the compassionate attitude of Bodhicitta, not really taking in anyone’s suffering.
If it is possible to suffer on behalf of another, since there are immeasurable Buddhas with perfect compassion and meritorious virtues, they would have removed the suffering of all beings instantly long ago. However, there are immeasurable beings still suffering.
The most skilful way Buddhas can help beings in suffering is to urge them to be mindful of their names sincerely, so as to rely on their meritorious virtues to eradicate their negative karma, for departing from suffering. Note that is still not the Buddhas suffering on behalf of anyone, but offering everyone a path to end their suffering. (However, before they became Buddhas, they did suffer on the path of practice, for the sake of benefitting many.)
On our part, other than encouraging those suffering to practise the above to alleviate suffering, we can also practise repentance, mindfulness of Buddha and dedication of merits on the behalf of all concerned. This works to some extent as there is collective karma connecting those involved. Of course, it is best that those suffering sincerely practise likewise too, for direct benefits.