Is it permissible for a woman observing iddah to communicate with a relative of her deceased husband in writing via the internet to inquire about her well-being and her daughter's?

This woman is a non-mahram, so she must be treated as such. Islamic law does not permit communication with her except to the extent of need and with the assurance that there will be no temptation (fitna). If it is possible to marry her after her 'iddah (waiting period) expires, then do so as an act of kindness to her and her son.