Begin by reviewing the Aircall API documentation to understand the available endpoints and the data structure. Identify the specific data you need to extract, such as call logs, user data, or call statistics. Ensure you have access to the Aircall API with the necessary credentials and permissions.

Use the API credentials obtained from Aircall to authenticate your requests. Aircall typically uses API keys or OAuth tokens for authentication. Make sure you securely store these credentials as they will be required for making API requests.

Write a script or use a command-line tool to make HTTP GET requests to the relevant Aircall API endpoints. Ensure you handle pagination if the data is too large to be retrieved in a single request. Parse the JSON responses and extract the required data fields. Save this data into a local file, such as a CSV or JSON file, for temporary storage.

Clean and transform the extracted data to match the schema of your Snowflake database. This may include renaming fields, converting data types, or normalizing data. Save the processed data into a CSV file format, as this is widely supported and can easily be ingested by Snowflake.

Log in to your Snowflake account and create a stage to store the data files temporarily before loading them into a table. You can create an internal stage using the following SQL command:
```sql
CREATE STAGE my_aircall_stage;
```

Use the Snowflake web interface or the SnowSQL command-line tool to upload your prepared CSV data file to the Snowflake stage created in the previous step. For example, using SnowSQL, you can execute:
```bash
snowsql -q "PUT file://path/to/your/datafile.csv @my_aircall_stage;"
```

Create a table in Snowflake that matches the structure of your CSV file. Use the `COPY INTO` command to load data from the staged file into the Snowflake table. Here is an example command:
```sql
CREATE TABLE aircall_data (
column1 STRING,
column2 STRING,
...
);

COPY INTO aircall_data
FROM @my_aircall_stage/datafile.csv
FILE_FORMAT = (TYPE = 'CSV' FIELD_OPTIONALLY_ENCLOSED_BY = '"' SKIP_HEADER = 1);
```

By following these steps, you can successfully move data from Aircall to Snowflake Data Cloud without using third-party connectors or integrations.