Yes having a blood test done can detect if there's breast cancer or other forms of cancer inside your body.
The blood test can also test to make sure your other organs such as kidneys are functioning properly as a result of cancer.
You still should have a mammogram done though to fully detect the breast cancer but blood tests can be the first thing that shows whether you may have breast cancer or not.
If breast cancer is caught soon enough and treated soon enough then you have a much higher chance of survival.