To store the quantity of heat required without rejecting any of it is a big ask. As a quick feasibility check, @ ~6 liters/100km I use ~ 1 liter of fuel each daily trip. If we make the task simpler by only capturing heat from the cooling system and assume 43MJ/kg, 0.76kg/l and 30% of that heat into the cooling system, 16.97MJ will have to be absorbed by the storage system. It is reasonable to assume that @ 3l/100km, that goes to 0.5l and 8.49MJ:
Barium hydroxide has been used for latent heat storage in vehicles so starting with that:
From here:
Salt Hydrates - Melting points and Latent Energy
we need 8.49 (MJ)/6.55 (MJ/m^3) = 1.30 m^3 = 1300l of Ba(OH)2.8(H2O). It has to be insulated (vacuum?), paid for, will weigh 4.85 metric tons (just the Ba(OH)~ )and some provision for transferring heat to and from it made. Perhaps not.
A lesser quantity of stored heat might work, depending on what is the heat energy required to heat the house.
You might use something else to store the heat. Water to steam (2257kJ/kg) comes to mind, but then you need a pressure vessel, and maintaining the required temperature becomes an issue. (Paraffin wax has a latent heat of 147kJ/kg while Ba(OH).8H2O has 174kJ/kg, - and the heat is stored at a more useful temperature; 78C vs 47C.)
What will work is using stored heat to allow faster engine warm up. Cold start enrichment has a strong effect on fuel consumption, the shorter the trip, the greater the effect.
There used to be commercially available units to store cooling system heat but the units used in the various Prius models will be easier to find. As used items, less expensive too. (Maybe someone has already plumbed such a unit in and posted on here with the results?)