Add an incremental number in a field in INSERT INTO SELECT query in SQL Server

You can use the row_number() function for this.

INSERT INTO PM_Ingrediants_Arrangements_Temp(AdminID, ArrangementID, IngrediantID, Sequence)
    SELECT @AdminID, @ArrangementID, PM_Ingrediants.ID,
            row_number() over (order by (select NULL))
    FROM PM_Ingrediants 
    WHERE PM_Ingrediants.ID IN (SELECT ID FROM GetIDsTableFromIDsList(@IngrediantsIDs)
                             )

If you want to start with the maximum already in the table then do:

INSERT INTO PM_Ingrediants_Arrangements_Temp(AdminID, ArrangementID, IngrediantID, Sequence)
    SELECT @AdminID, @ArrangementID, PM_Ingrediants.ID,
           coalesce(const.maxs, 0) + row_number() over (order by (select NULL))
    FROM PM_Ingrediants cross join
         (select max(sequence) as maxs from PM_Ingrediants_Arrangement_Temp) const
    WHERE PM_Ingrediants.ID IN (SELECT ID FROM GetIDsTableFromIDsList(@IngrediantsIDs)
                             )

Finally, you can just make the sequence column an auto-incrementing identity column. This saves the need to increment it each time:

create table PM_Ingrediants_Arrangement_Temp ( . . .
    sequence int identity(1, 1) -- and might consider making this a primary key too
    . . .
)