Essentially what I'm asking is that if someone's arm is amputated, doesn't that cut off a significant part of their blood flow because the blood vessels that normally flowed up to the hand of the amputated arm and back can't anymore? If it heals, does that mean the body has to grow more vessels to sort of complete the circuit again? Does the body just compensate by putting more stress on the rest of the body's blood vessels?
If I understand your question correctly, you ask how does the body deals with released blood volume after the removal of part of blood vessels?
The simple answer is the body decreases the blood volume. Our body is able to alter blood volume significantly in response to different factors.
For example, take a look at this paper: Blood volume changes in normal pregnancy. It states that:
A healthy woman bearing a normal sized fetus, with an average birth weight of about 3.3 kg, will increase her plasma volume by an average of about 1250 ml, a little under 50% of the average non-pregnant volume for white European women of about 2600 ml.
So, after the arm cutting, an organism will decrease its blood volume. Although it will take some time, blood volume altering is more probable way than new vessels generation.
Amputation of a limb or more will cut off both the arteries and veins to and from that limb. The body will compensate by reducing the amount of the circulating blood in it in order to maintain normal blood pressure. The best analogy I can provide would be electrical circuits: In amputation you are removing a circuit that is connected as a parallel circuit, not as a series circuit, therefor there is no need to create new connection after amputation to allow the circuit to complete, you just need to block the loose ends (ligate the main artery and vein to the amputated limb).
Amputation does not lead to increased stress on the remaining part of the system, in fact it has the exact opposite effect. The heart as a pump becomes more efficient as the "piping system" is now shorter with less resistance, and people with cardiac failure report feeling improved cardiac symptoms after amputation, and those with high blood pressure require less medication to control their blood pressure. In fact, the functional efficiency of all other vital organs like the kidneys, pancreas and liver improve as a result of reduced body mass (they now need to service and detox less cells). If the person was diabetic, their requirements for external insulin reduce, and people who suffer kidney failure might no longer require dialysis as the remaining kidney function -although compromised- becomes sufficient to filter the smaller amount of blood which includes less -waste chemicals- due to reduced muscle and fat mass.