If there’s no will, who gets the stuff in a rented house

There are two different issues here:
1) Who inherits? If there is no will, the items pass according to intestate succession, which are the rules for who gets what when there there is no will. If your mother-in-law was not married when she passed away, each living children of hers gets an equal share of the items (or their value, if they are sold); if any of her children pre-deceased (died before) her, then *their* children (if any) will share or divide the share their deceased parent would have received had he or she lived. So certainly, your husband is, depending on how many siblings and half-siblings he had and whether any of them passed away leaving neices or nephews of his, entitled to at least a share and possibly the entirety (e.g. if he was the only child of your mother-in-law).
2) However, being entitled to possessions does NOT let you enter another person's home without their permission. If the granddaughter owned the home and he went into the home without her permission, then at a minimum, he committed trespassing; he may have committed breaking and entering if the home had been locked at the time, and may also have committed theft if he took anything, since even if he will inherit some or all of the possession in the future, until they are properly distributed, they belong to her estate, not him. You can't simply go into a home and take things, even if you believe they are yours.