There are several payment processors available that can handle your pricing structure efficiently. Here are two popular options:

1. Stripe:
Stripe is a widely used payment processor known for its developer-friendly solutions. With Stripe, you can easily set up a subscription-based pricing model with a trial period. Here's how you can achieve your desired pricing structure using Stripe:

- Set up a subscription plan with a 90-day trial period using the Stripe API or dashboard.
- Define the plan's price as $0.01 for the trial period.
- After the trial period ends, update the subscription plan to charge $9.99 per month.

Stripe provides comprehensive documentation and code examples to help you implement these steps. You can refer to their official documentation on creating subscriptions (https://stripe.com/docs/billing/subscriptions/overview) and managing subscription pricing (https://stripe.com/docs/billing/subscriptions/prices) for detailed instructions.

2. PayPal:
PayPal is another popular payment processor that offers subscription management capabilities. With PayPal, you can create a subscription plan that includes a trial period and then switches to a monthly fee. Here's how you can set it up:

- Create a subscription plan using PayPal's API or the PayPal Business dashboard.
- Specify the trial period as 90 days and set the trial price to $0.01.
- After the trial period, update the subscription plan to charge $9.99 per month.

PayPal provides detailed documentation on creating subscription plans (https://developer.paypal.com/docs/subscriptions/) and managing subscription pricing (https://developer.paypal.com/docs/subscriptions/integrate/pricing/) that can guide you through the process.

Both Stripe and PayPal offer robust APIs, easy-to-use dashboards, and extensive documentation to help you integrate their services into your business. It's recommended to review their features, pricing, and terms to determine which processor best suits your specific needs and preferences.