You accept the price and agree to give them a downpayment, let's use 10% for this exercise, of $10,000 with the remaining $90,000 on a Promissory Note that is collateralized by a Deed of Trust on the subject property.

There are many ways to make this kind of purchase without using your own capital...seller financing, private money and angels are all great options.2) Type of loan and terms is contingent on lending entity criteria, deal parameters, buyer qualifications, collateral value, et al.  

A marginal property with no money down if you have cross collateral of other assets or a personal guarantee then you stand to lose a lot on the debt obligation.Properties like that can needs ongoing repairs.

Can I do a land contract for the deed and finance the down payment to the owner with a secure line of credit collateralized by the property?

I have done killer deals where I had to enforce my contract.One way to protect yourself is to have the other partner pledge some collateral to protect your investment.

Once a property owner becomes delinquent in payments, the lenders start to become more proactive in preservation of the collateral securing their loans.

These are asset-based loans, secured by the collateral.

In addition, many smaller banks allow you to use your cash as collateral.

You may want to try a smaller local bank, a portfolio lender, or you might be able to get other secured financing using the house as collateral (non mortgage loan). 

The rule of thumb I use is that if vacancy status can be established, we as lenders we have the right to protect our vested interest by taking actions to secure and preserve the collateral.