Empiricism and reason
'Empiricism' has as many meanings as there are empiricists. But if we take empiricism as the view that all knowledge derives ultimately from sense experience, which has some claims to be the standard view, there is no inconsistency in recognising a role for reason. We can reason about what we derive from experience. For instance, without experience we would not know what a colour is or know that red, green and blue are colours. However, given our experience of colours we can deduce that if X is red then X is coloured. We can recognise logical relations between concepts, in other words.
Whether induction is rational, non-rational or irrational is a matter of much dispute, not least among empiricists themselves, but we don't need to decide anything here. The red/ colour example is enough to show that the ultimate derivation of all knowledge from sense experience is perfectly consistent with the application of reason to what we derive from experience.
Empiricism and irrationalism
Different views may be taken about what irrationalism is but to help fix ideas:
'Irrationalism' often connotes a tendency toward or advocacy of arbitrariness
in one's beliefs and decisions. The thought is that an irrationalist is someone who
simply plumps for a particular belief or course of action without considering
reasons for or against it. An irrationalist is a person prone to 'leaps' rather than to
inferences. ... Another idea closely associated with arbitrariness is that 'irrationalism' is a view
that substitutes private whim or wishful thinking for rational conviction. That is,
an irrationalist cares more about what she wants to be true than what reason
Irrationalism also suggests a sort of denigration of reason. The idea is that the
deliverances of reason are irrelevant, immaterial, or worthless. Put in religious
terms, the thought is that human reason is limited and corrupt and so has no
rights over divine revelation. Allied with this denigration of reason is an aversion
to critical enquiry, particularly when it is directed at one's cherished beliefs. This
is often joined with a kind of counter-evidentialism. Counter-evidentialism goes
beyond the claim that some beliefs or courses of action can be justified even
when evidence is weak or absent to endorse the claim that they can justified even
if the evidence against them is overwhelming. Another version of this counsels
the adoption of beliefs or courses of action that are manifestly incoherent. This is
captured by the famous misquote from Tertullian, credo quia absurdum ['I believe because it is absurd'].
(Benjamin D. Crowe, 'F. H. Jacobi on Faith, or What It Takes to Be an Irrationalist', Religious Studies, Vol. 45, No. 3 (Sep., 2009), pp. 309-324: 311.)
These characterisations fit well with your statement:
'...irrationalists think we can't use reason to make conclusions'.
Two comments to connect back to empiricism:
(1) If empiricism derives all knowledge ultimately from sense experience it seems to be at odds with irrationalism as 'simply plumping for a belief' from 'private whim or wishful thinking'. Empirical knowledge is controlled by experience; irrationalism pays no or little heed to experience.
(2) Empiricism has no truck with irrationalist counter-evidentialism. It precisely relies on evidence - evidence derived ultimately from experience.
Empiricism and intuition
You mention intuition, so you want some account, not given so far, of intuition. This term, like empiricism itself, is open to a broad variety of meanings. My own understanding of intuition is that if I know something intuitively, I know it by a direct intellectual awareness not derived from sense experience. It follows that intuition - intuitive knowledge - is not empirical knowledge. Empiricism excludes intuition.
Intuition or intuitive knowledge is not rational knowledge either. Intuition involves 'instantaneous insight' (B. Williams, Descartes, London: Routledge, 2005: 73) not preceded by rational processes.
This does not connect intuition with the irrational, however. Intuition by its nature - inherently - delivers truth. It is 'instantaneous insight' into the truth. Irrationalism by contrast involves - endorses or promotes - intellectual arbitrariness, private whim and wishful thinking, none of which can deliver truth, an essential element in knowledge, other than accidentally.
Whether we actually have any intuitive knowledge as defined I cannot say.
I have not defended empiricism here and would not call myself an empiricist. I have simply answered your question on lines which I believe are open to an empiricist on what I have called the standard view.