As long as the part is designed for the appliance being repaired, it is possible to use any brand. Some brands of small appliances, including Sylvania and Magnavox, use parts that are compatible with their Emerson counterpart.
Any brand of light bulb is fine to use in an Emerson microwave oven as long as it has the correct type of base and electrical rating. Home Depot sells the Phillips brand, as of 2015.
A more elaborate repair, such as the heating element on an Emerson toaster oven, can use a non-original equipment part as long as it has the correct electrical rating and shape, but since a repair of this nature has serious risks of electrical shock and fire if done improperly, there is little to be gained by using a non-Emerson approved part.