In the event of a company's bankruptcy, how does a SPV continue operations if it can no longer collect payments on collateral Let's say a company sells car loans (or mortgages or credit cards, I don't think it really matters). The company sells the loans to an SPV, who then issues securities backed by the cash flows the SPV receives from the loan repayments, which, as far as I understand, come through the initial company (as it is the entity that services the car loans and repossesses them in the event that a car purchaser cannot make payments). 

If the company goes bankrupt, and must discontinue operations, I understand that the SPV is legally separate, and as such the company's creditors have no claim on the assets held by the SPV.

But how is the SPV going to collect/ repossess the car loans/ cars? It's not set up to do that. Would the SPV not also end up defaulting on its obligations at this point?
Servicing and origination don't have to be done by the same firm. It is quite common that the financing company contracts out the servicing rights on the loans they originate. 
Servicers, like many other kinds of firms, can continue to operate after bankruptcy. Bankruptcy is what happens when a firm can't pay its bills which includes debt, it doesn't mean that the firm needs to be liquidated. For example, existing debt holders are not getting paid and sue to put the firm into bankruptcy. The equity holders are wiped out, the debt holders become the new equity holders, and with the reduced debt costs the firm can often cover its costs and be profitable (even though the debt holders took a loss on their investment). 
Servicing can be reassigned after servicer bankruptcy. In fact, not only does this happen but it is common for a servicer to change when a loan is sold. So transfer of servicing rights is commonplace.