Yes, assuming that he is not too old. The age would depend on decision of the bishop and the individual’s circumstances. Generally 45-50 would be a cut-off. If the guy is 50 and has no college education, then it’s going to be quite hard to find any place to accept him. If he is 55 and completed all or most of seminary, they may be more inclined to consider him. If he is a deacon at 60, then even then they might consider it.
There are no set rules for this. It all depends on the man and the diocese he is interested in. Nevertheless, in short, you can be a widower and become a priest in the Latin Rite. This is keeping in mind that eastern rites have married priests and there are exceptions for married former protestant ministers within the Latin Rite.
If he has children, then they need to be self-sufficient adults. Again, this is not an absolute rule, but it would be very unlikely for an exception to be made. An exception might and I stress might be made if he had a mentally or physically challenged adult child.
In terms of having one or more young children there is pretty much no way he could enter seminary until they are fully grown. It might, might be possible if he had money to support them until adulthood and lived near a seminary and both the seminary and bishop were okay with him going to seminary during the day and returning home at night.
In these cases, it would be very unlikely, but it is always possible.
In short:
Widower: Fine, assuming no children or self-sufficient adult children. Age and education would be factors.
Widower with young children: Extremely unlikely but not impossible.