I have serious doubts that most state-run health or private insurance plans, or most hospitals or clinics, would approve of such self-administration.
First, IVIG is very expensive, so whoever is paying for it usually wants to ensure that (1) it was indeed infused and properly so; (2) it was infused to the right person -- their patient/insured, and not resold to someone else; and (3) it was infused in a monitored manner without causing harm to the patient/insured -- and a blood product like IVIG can definitely cause allergic reactions or troubling changes in blood pressure (because of the fluid volume/speed), or other side effects that could endanger the patient/insured. There are potentially serious liability issues for whoever sends a patient home with such medication usually not intended or approved for self-administration.
Second, in many jurisdictions there are policies against, or guidelines about, the discharge of patients from hospitals or clinics with IV lines or other catheters inside them, because they could be prone to abuse (e.g., to inject other drugs) or cause infections if not properly cleaned, or they could even lead to a hemorrhage if not properly removed. These are also potentially serious liability issues.
Third, beyond the fact that the IVIG must usually be measured and mixed in sterile conditions -- namely, it must be customized -- for a specific patient, and it must be temperature-controlled and the like, there also are practical risks that the IVIG container could break -- sometimes it comes in glass containers -- or that it could leak or develop air bubbles or other infusion-related issues which a patient may not be able to handle safely on his/her own.
To give a concrete example, when I get my IVIG at a hospital infusion clinic in Washington DC on Day #1, they let me go home with the IV still inside me on an overnight basis -- properly protected and wrapped, of course -- such that it can be used again on Day #2.
However, when I get my IVIG at a hospital infusion clinic in New York City on Day #1, they remove the IV at the end and thus they have to insert a new one on Day #2. The reason is a municipal public health regulation that forbids medical centers from discharging patients with IV lines in them, out of concern for what could happen once the patient leaves the controlled medical environment. And hospitals and clinics there like it that way, of course, because they don't have to worry about things going wrong -- and their being liable -- for whatever happens once as patient leaves their care.
We do have some patients in this community who get their IVIG infusions at home, but under the supervision of a visiting nurse. In one case, a patient is given his infusion at home by his wife, but she happens to be a nurse and the convenient arrangement was approved by all the parties concerned.