One solution would be to create infinite loop (while True) and break out of it when we encounter word "end"
vowels = 'aeiouAEIOU'
while True:
    text = input()  # it will get a new value every time the loop executes
    count = 0  # note it'll be assigned 0 every iteration
    if text == "end":
       break  # end the loop, since we found the word "end"
    # if we get here, it means the word is not "end"
    for letter in text:
        if letter in vowels:  # letter is a vowel, increase count by 1
            count += 1
    print(count)
To first read all the text and only then count/print the result you could first collect all the strings in a list.
vowels = 'aeiouAEIOU'
user_text_list = []  # we create empty list
while True:
    text = input()  # get input from user
    if text == "end":
       break  # end the loop, since we found the word "end"
    # if we get here, it means the word is not "end"
    user_text_list.append(text)  # "save" the word in a list
# once we get here, it means we encountered the word "end" and we can count the vowels
for word in user_text_list:  # for each element of the user_text_list (user inputted word)
    count = 0  # initialize count to 0 every iteration
    for letter in word:
        if letter in vowels:  # letter is a vowel, increase count by 1
            count += 1
    print(word, count)  # print word and vowel count
There isn't really an easy way to do it. Python waits for user to press [ENTER] when using input. If that's what you want, please refer to this question.
wordlist = []
while True:
    text = input()
    text = text.split(" ") # converts "I LOVE YOU" to ["I", "LOVE", "YOU"]
    for word in text:
       if word == "end":
           break
       wordlist.append(word)  # if word is not "end", add word to list
for word in wordlist:  # for each element of the user_text_list (user inputted word)
    count = 0  # initialize count to 0 every iteration
    for letter in word:
        if letter in vowels:  # letter is a vowel, increase count by 1
            count += 1
    print(word, count)  # print word and vowel count