Thinking Machines serves Inkling through Tinker, its training platform, which exposes Anthropic- and OpenAI-compatible endpoints. Tinker's docs describe them as a beta meant for testing models, not high-traffic production.
Create a Tinker API key in the Tinker console and send it as the x-api-key header (a Bearer header also works). In the examples it's written {{TINKER_API_KEY}}.
Full reference: https://tinker-docs.thinkingmachines.ai/tinker/compatible-apis/anthropic/
Example from Tinker's docs.
curl https://tinker.thinkingmachines.dev/services/tinker-prod/anthropic/api/v1/messages \
-H "x-api-key: {{TINKER_API_KEY}}" \
-H "content-type: application/json" \
-d '{"model": "thinkingmachines/Inkling", "max_tokens": 512, "messages": [{"role": "user", "content": "The capital of France is"}]}'