In principle you will need a valid and licensed ArcGIS installation to write to an ArcSDE Geodatabase.
But in practice it depends on what you need to write. If you only need to write attributes, you can use the native writer for the underlying SDE database, e.g. SQL Server.
If you need to write geometries it is slightly more complicated, but it depends on your geometries and how they are stored in the database. If your geometries are stored as spatial objects native to your database (e.g. SQL Server Spatial), you can probably use that writer to write directly into the table on which your feature class is based. Just be aware that any SDE specific attributes such as extents may or may not automatically be updated. Important: If you're inserting data you will also have to make sure to use the SDE-based generator for OBJECTID values, otherwise you may experience collisions later, see http://desktop.arcgis.com/en/arcmap/10.3/manage-data/using-sql-with-gdbs/next-rowid.htm
If you need to write data into SDE-specific datasets, e.g. versioned tables, it's going to be a lot more complicated/not feasible.
If you can tell us a bit more about your exact requirements it would be easier to give more concrete advice.
