Here are the likely culprits:
* Vacuum Leak: This is the most common cause. A leak in the vacuum lines connecting the engine to the brake booster will reduce or eliminate the assist. Check all the hoses and connections for cracks, holes, or loose clamps. Pay particular attention to the hose connecting the intake manifold to the brake booster.
* Brake Booster Failure: The brake booster itself could be failing. It's a diaphragm-type device that uses vacuum to amplify pedal pressure. A leak in the diaphragm or a malfunctioning valve would result in a soft pedal. This usually requires replacement of the booster.
* Master Cylinder Issues (Less Likely): While less likely to cause a pedal that's *only* soft when the engine is running, a failing master cylinder *could* contribute if it's already low on fluid or has internal leaks. Check the brake fluid level in the master cylinder reservoir. Low fluid points to a leak somewhere in the system (e.g., in a wheel cylinder or caliper).
* Restricted Airflow to the Intake Manifold (Rare): A severely restricted air intake could reduce the vacuum available to the booster, but this is less common and usually accompanied by other drivability issues.
Troubleshooting Steps:
1. Check Brake Fluid Level: Make sure the brake fluid level in the master cylinder reservoir is within the normal range. Low fluid is a serious issue that needs immediate attention.
2. Inspect Vacuum Lines: Carefully examine all vacuum lines connected to the brake booster. Look for cracks, holes, or loose connections. Listen for hissing sounds when the engine is running, indicating a leak.
3. Listen for Leaks: With the engine running, have someone pump the brake pedal several times and then hold it down. Listen carefully near the brake booster and vacuum lines for any hissing sounds indicative of a leak.
4. Test the Booster (Difficult without specialized tools): A definitive booster test often requires a vacuum pump to check for proper diaphragm function. This is best left to a mechanic.
If you're not comfortable with these checks, it's best to take your Corolla to a qualified mechanic for diagnosis and repair. Driving with a soft brake pedal is unsafe.