Try: exceeded, beekeeper, defenselessness, effervescence, enfeeblement, freebee, referee, teepee, equestrienne. Some of those have more than 4, hope that's okay.
Some four letter words that start with E are:eachearneaseeasteasyechoecruedgeeditelseemitenvyepicetcheuroeveneverevilexamexit
According to SOWPODS (the combination of Scrabble dictionaries used around the world) there are 1 words with the pattern E-EE. That is, four letter words with 1st letter E and 3rd letter E and 4th letter E. In alphabetical order, they are: epee
Four letter words with o e in the middle are: coeddoerdoesfoesgoergoeshoedhoerhoesJoeljoesjoeykoelnoelnoespoempoetroestoeatoedtoesvoeswoeszoea
Some four letter words that have both A and E in them:acreaideareabakebalebarebasebeadbeambeanbeatcafecakecamecanecapecarecasedaledamedaredatedeaddealdeandeareachearlearneaseeasteasyelanfacefadefakefamefarefatefazefearfeatfleagalegamegapegategazegearharehatehavehazeheadhealheaphearheatideajadelaceladelakelamelanelatelazeleadleafleakleanleapmacemademakemalemanemarematemazemeadmealmeanmeatnamenapenearneatpacepagepalepaneparepatepavepeakpealpearpeatplearaceragerakeraperarerateraverazereadrealreamreaprearsafesagesakesalesamesanesatesavesealseamsearseattaketaletametapetareteaktealteamtearteatureavalevanevasevealwadewagewakewanewarewaveweakwealweanwearyearzeal
According to SOWPODS (the combination of Scrabble dictionaries used around the world) there are 1 words with the pattern EU-E. That is, four letter words with 1st letter E and 2nd letter U and 4th letter E. In alphabetical order, they are: euge
According to SOWPODS (the combination of Scrabble dictionaries used around the world) there are 2 words with the pattern WE-E. That is, four letter words with 1st letter W and 2nd letter E and 4th letter E. In alphabetical order, they are: were wexe
According to SOWPODS (the combination of Scrabble dictionaries used around the world) there are 1 words with the pattern E-EC. That is, four letter words with 1st letter E and 3rd letter E and 4th letter C. In alphabetical order, they are: exec
According to SOWPODS (the combination of Scrabble dictionaries used around the world) there are 1 words with the pattern E-ET. That is, four letter words with 1st letter E and 3rd letter E and 4th letter T. In alphabetical order, they are: evet
According to SOWPODS (the combination of Scrabble dictionaries used around the world) there are 1 words with the pattern E-EU. That is, four letter words with 1st letter E and 3rd letter E and 4th letter U. In alphabetical order, they are: emeu
According to SOWPODS (the combination of Scrabble dictionaries used around the world) there are 1 words with the pattern E-EV. That is, four letter words with 1st letter E and 3rd letter E and 4th letter V. In alphabetical order, they are: erev
According to SOWPODS (the combination of Scrabble dictionaries used around the world) there are 1 words with the pattern E-EW. That is, four letter words with 1st letter E and 3rd letter E and 4th letter W. In alphabetical order, they are: enew
According to SOWPODS (the combination of Scrabble dictionaries used around the world) there are 3 words with the pattern EA-E. That is, four letter words with 1st letter E and 2nd letter A and 4th letter E. In alphabetical order, they are: eale ease eave