The code you've shown in your question likely worked to save the vectors.
You can check the class type of the object that the api.load() gave you with code like:
print(type(vectors))
I think that will be shown to be an instance of KeyedVectors.
If so, then reloading the file you've already saved should be as simple as:
from gensim.models import KeyedVectors
reloaded_vectors = KeyedVectors.load('vectors.bin')
Note that the native Gensim .save(), for any vectors of significant size, will usually be split over multiple files that all start with vectors.bin – so if you then want to move the model elsewhere, move all those files together.
You can also save/load vectors in a simpler format with the .save_word2vec_format() & .load_word2vec_format() methods. EG:
vectors.save_word2vec_format('vectors.txt', binary=False)
vectors_reloaded = KeyedVectors.load_word2vec_format('vectors.txt', binary=False)