Is it possible to reproject spatial data using SQL Server?

No.

Transform - ability to transform from one spatial ref to another: No - need 3rd-party tools, Geometry can use any SRID between 0 and 999999. Spatial Tools free CLR add-on does provide limited transform support.

Source: http://www.bostongis.com/PrinterFriendly.aspx?content_name=sqlserver2008r2_oracle11gr2_postgis15_compare

Not by default, but check the SQL Server Spatial Tools developed by MSDN on GitHub. Specifically, the affine transformation functions.

Example for reprojection from EPSG:2193 to EPSG:3857

c:\OSGeo4W64\bin\ogr2ogr.exe ^
 -f "MSSQLSpatial"^
 "MSSQL:server=DestServerName;database=DestDbName;trusted_connection=yes"^
 "MSSQL:server=SourceServerName;database=SourceDbName;trusted_connection=yes"^
 -sql "SELECT [Id], [Shape].STAsText() Shape FROM [SourceDbName].[dbo].[SourceTableName]"^
 -nln "DestTableName"^
 -overwrite^
 -s_srs EPSG:2193^
 -t_srs EPSG:3857

After this execute SQL Query

update  [DestDbName].[dbo].[DestTableName]
set     [ogr_geometry] =  geometry::STGeomFromText([shape], 3857)