Not unless $\delta$ has countable cofinality (e.g., $\delta = \aleph_\omega$). This will fail for $\delta = \aleph_1$, for example. Let $\phi: \delta \to \delta$ be any increasing function and recursively define $\lambda_0 = 0$ and $\lambda_{n+1} = \phi(\lambda_n)+1$. Since $\phi$ is increasing, the sequence $(\lambda_n)$ is increasing, and since $\delta$ has uncountable cofinality we have $\lambda = \sup \lambda_n < \delta$. However, for any $\gamma < \lambda$ we must have $\gamma < \lambda_n$ for some $n$, so that $\phi(\gamma) < \leq \lambda_{n+1} < \lambda$.
(If $\delta$ has countable cofinality it's easy. For instance, if $\delta = \aleph_\omega$ then we define $\phi$ by letting $\phi(\lambda) = \aleph_{n+1} + \lambda$ for $\aleph_n \leq \lambda < \aleph_{n+1}$.)
Not unless $\delta$ has countable cofinality (e.g., $\delta = \aleph_\omega$). This will fail for $\delta = \aleph_1$, for example. Let $\phi: \delta \to \delta$ be any increasing function and recursively define $\lambda_0 = 0$ and $\lambda_{n+1} = \phi(\lambda_n)+1$. Since $\phi$ is increasing, the sequence $(\lambda_n)$ is increasing, and since $\delta$ has uncountable cofinality we have $\lambda = \sup \lambda_n < \delta$. However, for any $\gamma < \lambda$ we must have $\gamma < \lambda_n$ for some $n$, so that $\phi(\gamma) < \lambda_{n+1} < \lambda$.
(If $\delta$ has countable cofinality it's easy. For instance, if $\delta = \aleph_\omega$ then we define $\phi$ by letting $\phi(\lambda) = \aleph_{n+1} + \lambda$ for $\aleph_n \leq \lambda < \aleph_{n+1}$.)